Theory Of Games And Economic Behavior

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THEORY OF GAMES
AND ECONOMIC BEHAVIOR

THEORY OF
GAMES

AND ECONOMIC
BEHAVIOR

By JOHN VON NEUMANN, and
OSKAR MORGENSTERN

PRINCETON

PRINCETON UNIVERSITY PRESS
1953

Copyright l9kU, by Princeton University Press

PRINTED IN THE UNITED STATES OF AMERICA

Second printing (second edition), 1947
Third printing, 1948
Fourth printing, 1950
Fifth printing (third edition), 1953
Sixth printing, 1955

London: Geoffrey Cumberlege • Oxford University Press

PREFACE TO FIRST EDITION

This book contains an exposition and various applications of a mathe-
matical theory of games. The theory has been developed by one of us
since 1928 and is now published for the first time in its entirety. The
applications are of two kinds : On the one hand to games in the proper sense,
on the other hand to economic and sociological problems which, as we hope
to show, are best approached from this direction.

The applications which we shall make to games serve at least as much
to corroborate the theory as to investigate these games. The nature of this
reciprocal relationship will become clear as the investigation proceeds.
Our major interest is, of course, in the economic and sociological direction.
Here we can approach only the simplest questions. However, these ques-
tions are of a fundamental character. Furthermore, our aim is primarily
to show that there is a rigorous approach to these subjects, involving, as
they do, questions of parallel or opposite interest, perfect or imperfect infor-
mation, free rational decision or chance influences.

John von Neumann
Oskar Morgenstern.

Princeton, N. J.

January, 1943.

PREFACE TO SECOND EDITION

The second edition differs from the first in some minor respects only.
We have carried out as complete an elimination of misprints as possible, and
wish to thank several readers who have helped us in that respect. We have
added an Appendix containing an axiomatic derivation of numerical utility.
This subject was discussed in considerable detail, but in the main qualita-
tively, in Section 3. A publication of this proof in a periodical was promised
in the first edition, but we found it more convenient to add it as an Appendix.
Various Appendices on applications to the theory of location of industries
and on questions of the four and five person games were also planned, but
had to be abandoned because of the pressure of other work.

Since publication of the first edition several papers dealing with the
subject matter of this book have appeared.

The attention of the mathematically interested reader may be drawn
to the following: A. Wald developed a new theory of the foundations of
statistical estimation which is closely related to, and draws on, the theory of

VI

PREFACE TO SECOND EDITION

the zero-sum two-person game (“ Statistical Decision Functions Which
Minimize the Maximum Risk,” Annals of Mathematics, Vol. 46 (1945)
pp. 265-280). He also extended the main theorem of the zero-sum two-
person games (cf. 17.6.) to certain continuous-infinite-cases, (“ Generalization
of a Theorem by von Neumann Concerning Zero-Sum Two-Person Games,”
Annals of Mathematics, Vol. 46 (1945), pp. 281-286.) A new, very simple
and elementary proof of this theorem (which covers also the more general
theorem referred to in footnote 1 on page 154) was given by L. H. Loomis ,
(“On a Theorem of von Neumann,” Proc. Nat. Acad., Vol. 32 (1946) pp. 213-
215). Further, interesting results concerning the role of pure and of mixed
strategies in the zero-sum two-person game were obtained by I. Kaplanski ,
(“A Contribution to von Neumann's Theory of Games,” Annals of Mathe-
matics, Vol. 46 (1945), pp. 474-479). We also intend to come back to vari-
ous mathematical aspects of this problem. The group theoretical problem
mentioned in footnote 1 on page 258 was solved by C. Chevalley.

The economically interested reader may find an easier approach to the
problems of this book in the expositions of L. Hurwicz , (“The Theory of
Economic Behavior,” American Economic Review, Vol. 35 (1945), pp. 909-
925) and of J. Marschak (“Neumann's and Morgenstern's New Approach
to Static Economics,” Journal of Political Economy, Vol. 54, (1946),
pp. 97-115).

John von Neumann
Oskar Morgenstern

Princeton, N. J.

September, 1946.

PREFACE TO THIRD EDITION

The Third Edition differs from the Second Edition only in the elimination
of such further misprints as have come to our attention in the meantime,
and we wish to thank several readers who have helped us in that respect.

Since the publication of the Second Edition, the literature on this subject
has increased very considerably. A complete bibliography at this writing
includes several hundred titles. We are therefore not attempting to give
one here. We will only list the following books on this subject:

(1) H. W . Kuhn and A. W . Tucker (eds.), “Contributions to the Theory
of Games, I,” Annals of Mathematics Studies, No. 24, Princeton (1950),
containing fifteen articles by thirteen authors.

(2) H. W. Kuhn and A. W. Tucker (eds.), “Contributions to the Theory
of Games, II,” Annals of Mathematics Studies, No. 28, Princeton (1953),
containing twenty-one articles by twenty-two authors.

(3) J. McDonald , Strategy in Poker, Business and War, New York
(1950).

(4) J. C. C. McKinsey , Introduction to the Theory of Games, New
York (1952).

(5) A. Wald , Statistical Decision Functions, New York (1950).

(6) J. Williams, The Compleat Strategyst, Being a Primer on the Theory
of Games of Strategy, New York (1953).

Bibliographies on the subject are found in all of the above books except
(6). Extensive work in this field has been done during the last years by the
staff of the RAND Corporation, Santa Monica, California. A bibliography
of this work can be found in the RAND publication RM-950.

In the theory of n- person games, there have been some further develop-
ments in the direction of “ non-cooperative ” games. In this respect,
particularly the work of J. F. Nash, “Non-cooperative Games,” Annals of
Mathematics, Yol. 54, (1951), pp. 286-295, must be mentioned. Further
references to this work are found in (1), (2), and (4).

Of various developments in economics we mention in particular “linear
programming” and the “assignment problem” which also appear to be
increasingly connected with the theory of games. The reader will find
indications of this $gain in (1), (2), and (4).

The theory of utility suggested in Section 1.3., and in the Appendix to the
Second Edition has undergone considerable development theoretically, as
well as experimentally, and in various discussions. In this connection, the
reader may consult in particular the following:

M. Friedman and L. J. Savage, “The Utility Analysis of Choices Involv-
ing Risk,” Journal of Political Economy, Vol. 56, (1948), pp. 279-304.

vii

viii

PREFACE TO THIRD EDITION

J. Marschak, “ Rational Behavior, Uncertain Prospects, and Measurable
Utility, ” Econometrica, Vol. 18, (1950), pp. 111-141.

F. Mosteller and P. Nogee , “ An Experimental Measurement of Utility,”
Journal of Political Economy, Vol. 59, (1951), pp. 371-404.

M. Friedman and L. J. Savage , “The Expected- Utility Hypothesis and
the Measurability of Utility, ” Journal of Political Economy, Vol. 60,
(1952), pp. 463-474.

See also the Symposium on Cardinal Utilities in Econometrica, Vol. 20,
(1952):

H. Wold , “Ordinal Preferences or Cardinal Utility?”

A. S. Manne , “The Strong Independence Assumption — Gasoline
Blends and Probability Mixtures.”

P. A. Samuelson, “Probability, Utility, and the Independence Axiom.”

E. Malinvaud , “Note on von Neumann-Morgenstern’s Strong Inde-
pendence Axiom.”

In connection with the methodological critique exercised by some of the
contributors to the last-mentioned symposium, we would like to mention
that we applied the axiomatic method in the customary way with the cus-
tomary precautions. Thus the strict, axiomatic treatment of the concept
of utility (in Section 3.6. and in the Appendix) is complemented by an
heuristic preparation (in Sections 3. 1.-3. 5.). The latter’s function is to
convey to the reader the viewpoints to evaluate and to circumscribe the
validity of the subsequent axiomatic procedure. In particular our dis-
cussion and selection of “natural operations” in those sections covers what
seems to us the relevant substrate of the Samuelson-Malinvaud “inde-
pendence axiom.”

John von Neumann
Oskar Morgenstern

Princeton, N. J.

January, 1953.

TECHNICAL NOTE

The nature of the problems investigated and the techniques employed
in this book necessitate a procedure which in many instances is thoroughly
mathematical. The mathematical devices used are elementary in the sense
that no advanced algebra, or calculus, etc., occurs. (With two, rather unim-
portant, exceptions: Part of the discussion of an example in 19.7. et sequ. and
a remark in A. 3. 3. make use of some simple integrals.) Concepts originating
in set theory, linear geometry and group theory play an important role, but
they are invariably taken from the early chapters of those disciplines and are
moreover analyzed and explained in special expository sections. Neverthe-
less the book is not truly elementary because the mathematical deductions
are frequently intricate and the logical possibilities are extensively exploited.

Thus no specific knowledge of any particular body of advanced mathe-
matics is required. However, the reader who wants to acquaint himself
more thoroughly with the subject expounded here, will have to familiarize
himself with the mathematical way of reasoning definitely beyond its
routine, primitive phases. The character of the procedures will be mostly
that of mathematical logics, set theory and functional analysis.

We have attempted to present the subject in such a form that a reader
who is moderately versed in mathematics can acquire the necessary practice
in the course of this study. We hope that we have not entirely failed in
this endeavour.

In accordance with this, the presentation is not what it would be in a
strictly mathematical treatise. All definitions and deductions are con-
siderably broader than they would be there. Besides, purely verbal dis-
cussions and analyses take up a considerable amount of space. We have
in particular tried to give, whenever possible, a parallel verbal exposition
for every major mathematical deduction. It is hoped that this procedure
will elucidate in unmathematical language what the mathematical technique
signifies — and will also show where it achieves more than can be done
without it.

In this, as well as in our methodological stand, we are trying to follow
the best examples of theoretical physics.

The reader who is not specifically interested in mathematics should at
first omit those sections of the book which in his judgment are too mathe-
matical. We prefer not to give a definite list of them, since this judgment
must necessarily be subjective. However, those sections marked with an
asterisk in the table of contents are most likely to occur to the average reader
in this connection. At any rate he will find that these omissions will little
interfere with the comprehension of the early parts, although the logical

TECHNICAL NOTE

chain may in the rigorous sense have suffered an interruption. As he
proceeds the omissions will gradually assume a more serious character and
the lacunae in the deduction will become more and more significant. The
reader is then advised to start again from the beginning since the greater
familiarity acquired is likely to facilitate a better understanding.

ACKNOWLEDGMENT

The authors wish to express their thanks to Princeton University and to
the Institute for Advanced Study for their generous help which rendered
this publication possible.

They are also greatly indebted to the Princeton University Press which
has made every effort to publish this book in spite of wartime difficulties.
The publisher has shown at all times the greatest understanding for the
authors 1 wishes.

CONTENTS

Preface v

Technical Note ix

Acknowledgment x

CHAPTER I

FORMULATION OF THE ECONOMIC PROBLEM

1. The Mathematical Method in Economics 1

1.1. Introductory remarks 1

1.2. Difficulties of the application of the mathematical method 2

1.3. Necessary limitations of the objectives 6

1.4. Concluding remarks 7

2. Qualitative Discussion of the Problem of Rational Behav-
ior 8

2.1. The problem of rational behavior 8

2.2. “Robinson Crusoe" economy and social exchange economy 9

2.3. The number of variables and the number of participants 12

2.4. The case of many participants: Free competition 13

2.5. The “Lausanne" theory 15

3. The Notion of Utility 15

3.1. Preferences and utilities 15

3.2. Principles of measurement: Preliminaries 16

3.3. Probability and numerical utilities 17

3.4. Principles of measurement: Detailed discussion 20

3.5. Conceptual structure of the axiomatic treatment of numerical

utilities 24

3.6. The axioms and their interpretation 26

3.7. General remarks concerning the axioms 28

3.8. The role of the concept of marginal utility 29

4. Structure of the Theory: Solutions and Standards of

Behavior 31

4.1. The simplest concept of a solution for one participant 31

4.2. Extension to all participants 33

4.3. The solution as a set of imputations 34

4.4. The intransitive notion of “superiority" or “domination" 37

4.5. The precise definition of a solution 39

4.6. Interpretation of our definition in terms of “standards of behavior" 40

4.7. Games and social organizations 43

4.8. Concluding remarks 43

CHAPTER II

GENERAL FORMAL DESCRIPTION OF GAMES OF STRATEGY

5. Introduction 46

5.1. Shift of emphasis from economics to games 46

5.2. General principles of classification and of procedure 46

CONTENTS

6. The Simplified Concept of a Game 48

6.1. Explanation of the termini iechnici 48

6.2. The elements of the game 49

6.3. Information and preliminary 51

6.4. Preliminarity, transitivity, and signaling 51

7. The Complete Concept of a Game 55

7.1. Variability of the characteristics of each move 55

7.2. The general description 57

8. Sets and Partitions 60

8.1. Desirability of a set-theoretical description of a game 60

8.2. Sets, their properties, and their graphical representation 61

8.3. Partitions, their properties, and their graphical representation 63

8.4. Logistic interpretation of sets and partitions 66

*9. The Set-theoretical Description of a Game 67

*9.1. The partitions which describe a game 67

*9.2. Discussion of these partitions and their properties 7 1

*10. Axiomatic Formulation 73

*10.1. The axioms and their interpretations 73

*10.2. Logistic discussion of the axioms 76

*10.3. General remarks concerning the axioms 76

*10.4. Graphical representation 77

11. Strategies and the Final Simplification of the Description

of a Game 79

11.1. The concept of a strategy and its formalization 79

11.2. The final simplification of the description of a game 81

11.3. The role of strategies in the simplified form of a game 84

11.4. The meaning of the zero-sum restriction 84

CHAPTER III

ZERO-SUM TWO-PERSON GAMES: THEORY

12. Preliminary Survey 85

12.1. General viewpoints 85

12.2. The one-person game 85

12.3. Chance and probability 87

12.4. The next objective 87

13. Functional Calculus 88

13.1. Basic definitions 88

13.2. The operations Max and Min 89

13.3. Commutativity questions 91

13.4. The mixed case. Saddle points 93

13.5. Proofs of the main facts 95

14. Strictly Determined Games 98

14 1. Formulation of the problem 98

14.2. The minorant and the majorant games 100

14.3. Discussion of the auxiliary games 101

CONTENTS

14.4. Conclusions 105

14.5. Analysis of strict determinateness 106

14.6. The interchange of players. Symmetry 109

14.7. Non strictly determined games 110

14.8. Program of a detailed analysis of strict determinateness 111

*15. Games with Perfect Information 112

*15.1. Statement of purpose. Induction 112

*15.2. The exact condition (First step) 114

*15.3. The exact condition (Entire induction) 116

*15.4. Exact discussion of the inductive step 117

*15.5. Exact discussion of the inductive step (Continuation) 120

*15.6. The result in the case of perfect information 123

*15.7. Application to Chess 124

*15.8. The alternative, verbal discussion 126

16. Linearity and Convexity 128

16.1. Geometrical background 128

16.2. Vector operations 129

16.3. The theorem of the supporting hyperplanes 134

16.4. The theorem of the alternative for matrices 138

17. Mixed Strategies. The Solution for All Games 143

17.1. Discussion of two elementary examples 143

17.2. Generalization of this viewpoint 145

17.3. Justification of the procedure as applied to an individual play 146

17.4. The minorant and the majorant games. (For mixed strategies) 149

17.5. General strict determinateness 150

17.6. Proof of the main theorem 153

17.7. Comparison of the treatment by pure and by mixed strategies 155

17.8. Analysis of general strict determinateness 158

17.9. Further characteristics of good strategies 160

17.10. Mistakes and their consequences. Permanent optimality 162

17.11. The interchange of players. Symmetry 165

CHAPTER IV

ZERO-SUM TWO-PERSON GAMES: EXAMPLES

18. Some Elementary Games 169

18.1. The simplest games 169

18.2. Detailed quantitative discussion of these games 170

18.3. Qualitative characterizations 173

18.4. Discussion of some specific games. (Generalized forms of Matching

Pennies) 175

18.5. Discussion of some slightly more complicated games 178

18.6. Chance and imperfect information 182

18.7. Interpretation of this result 185

*19. Poker and Bluffing 186

*19.1. Description of Poker 186

*19.2. Bluffing 188

*19.3. Description of Poker (Continued) 189

*19.4. Exact formulation of the rules 190

CONTENTS

*19.5. Description of the strategy 191

*19.6. Statement of the problem 195

*19.7. Passage from the discrete to the continuous problem 196

*19.8. Mathematical determination of the solution 199

*19.9. Detailed analysis of the solution 202

*19.10. Interpretation of the solution 204

*19.11. More general forms of Poker 207

*19.12. Discrete hands 208

*19.13. m possible bids 209

*19.14. Alternate bidding 211

*19.15. Mathematical description of all solutions 216

*19.16. Interpretation of the solutions. Conclusions 218

CHAPTER V

ZERO-SUM THREE-PERSON GAMES

20. Preliminary Survey 220

20.1. General viewpoints 220

20.2. Coalitions 221

21. The Simple Majority Game of Three Persons 222

21.1. Definition of the game 222

21.2. Analysis of the game: Necessity of “understandings’ 1 223

21.3. Analysis of the game: Coalitions. The role of symmetry 224

22. Further Examples 225

22.1. Unsymmetric distributions. Necessity of compensations 225

22.2. Coalitions of different strength. Discussion 227

22.3. An inequality. Formulae 229

23. The General Case 231

23.1. Detailed discussion. Inessential and essential games 231

23.2. Complete formulae 232

24. Discussion of an Objection 233

24.1. The case of perfect information and its significance 233

24.2. Detailed discussion. Necessity of compensations between three or

more players 235

CHAPTER VI

FORMULATION OF THE GENERAL THEORY:

ZERO-SUM n-PERSON GAMES

25. The Characteristic Function 238

25.1. Motivation and definition 238

25.2. Discussion of the concept 240

25.3. Fundamental properties 241

25.4. Immediate mathematical consequences 242

26. Construction of a Game with a Given Characteristic

Function 243

26.1. The construction 243

26.2. Summary 245

CONTENTS

27. Strategic Equivalence. Inessential and Essential Games 245

27.1. Strategic equivalence. The reduced form 245

27.2. Inequalities. The quantity y 248

27.3. Inessentiality and essentiality 249

27.4. Various criteria. Non additive utilities 250

27.5. The inequalities in the essential case 252

27.6. Vector operations on characteristic functions 253

28. Groups, Symmetry and Fairness 255

28.1. Permutations, their groups and their effect on a game 255

28.2. Symmetry and fairness 258

29. Reconsideration of the Zero-Sum Three-Person Game 260

29.1. Qualitative discussion 260

29.2. Quantitative discussion 262

30. The Exact Form of the General Definitions 263

30.1. The definitions 263

30.2. Discussion and recapitulation 265

*30.3. The concept of saturation 266

30.4. Three immediate objectives 271

31. First Consequences 272

31.1. Convexity, flatness, and some criteria for domination 272

31.2. The system of all imputations. One element solutions 277

31.3. The isomorphism which corresponds to strategic equivalence 281

32. Determination of All Solutions of the Essential Zero-Sum

Three-Person Game 282

32.1. Formulation of the mathematical problem. The graphical method 282

32.2. Determination of all solutions 285

33. Conclusions 288

33.1. The multiplicity of solutions. Discrimination and its meaning 288

33.2. Statics and dynamics 290

CHAPTER VII

ZERO-SUM FOUR-PERSON GAMES

34. Preliminary Survey 291

34.1. General viewpoints 291

34.2. Formalism of the essential zero sum four person games 291

34.3. Permutations of the players 294

35. Discussion of Some Special Points in the Cube Q 295

35.1. The corner /. (and F., F/., VII.) 295

35.2. The corner VIII. (and //., ///., /F.,). The three person game and

a “Dummy” 299

35.3. Some remarks concerning the interior of Q 302

36. Discussion of the Main Diagonals 304

36.1. The part adjacent to the corner VIII.' Heuristic discussion 304

36.2. The part adjacent to the corner VIII.: Exact discussion 307

*36.3. Other parts of the main diagonals 312

CONTENTS

37. The Center and Its Environs 313

37.1. First orientation about the conditions around the center 313

37.2. The two alternatives and the role of symmetry 315

37.3. The first alternative at the center 316

37.4. The second alternative at the center 317

37.5. Comparison of the two central solutions 318

37.6. Unsymmetrical central solutions 319

38. A Family op Solutions for a Neighborhood op the Center 321
*38.1. Transformation of the solution belonging to the first alternative at

the center 321

*38.2. Exact discussion 322

*38.3. Interpretation of the solutions 327

CHAPTER VIII

SOME REMARKS CONCERNING n ^ 5 PARTICIPANTS

39. The Number of Parameters in Various Classes of Games 330

39.1. The situation for n = 3, 4 330

39.2. The situation for all n ^ 3 330

40. The Symmetric Five Person Game 332

40.1. Formalism of the symmetric five person game 332

40.2. The two extreme cases 332

40.3. Connection between the symmetric five person game and the 1 , 2, 3-

8ymmetric four person game 334

CHAPTER IX

COMPOSITION AND DECOMPOSITION OF GAMES

41. Composition and Decomposition 339

41.1. Search for n- person games for which all solutions can be determined 339

41.2. The first type. Composition and decomposition 340

41.3. Exact definitions 341

41.4. Analysis of decomposability 343

41.5. Desirability of a modification 345

42. Modification of the Theory 345

42.1. No complete abandonment of the zero sum restriction 345

42.2. Strategic equivalence. Constant sum games 346

42.3. The characteristic function in the new theory 348

42.4. Imputations, domination, solutions in the new theory 350

42.5. Essentiality, inessentiality and decomposability in the new theory 351

43. The Decomposition Partition 353

43.1. Splitting sets. Constituents 353

43.2. Properties of the system of all splitting sets 353

43.3. Characterization of the system of all splitting sets. The decomposi-
tion partition 354

43.4. Properties of the decomposition partition 357

44. Decomposable Games. Further Extension of the Theory 358

44.1. Solutions of a (decomposable) game and solutions of its constituents 358

44.2. Composition and decomposition of imputations and of sets of impu-
tations 359

CONTENTS

44.3. Composition and decomposition of solutions. The main possibilities

and surmises 361

44.4. Extension of the theory. Outside sources 363

44.5. The excess 364

44.6. Limitations of the excess. The non-isolated character of a game in

the new setup 366

44.7. Discussion of the new setup. ^(e 0 ), F(e 0 ) 367

45. Limitations of the Excess. Structure of the Extended

Theory 368

45.1. The lower limit of the excess 368

45.2. The upper limit of the excess. Detached and fully detached imputa-
tions 369

45.3. Discussion of the two limits, |r|i, |r| 2 . Their ratio 372

45.4. Detached imputations and various solutions. The theorem con-
necting E(e 0 )j F(e 0 ) 375

45.5. Proof of the theorem 376

45.6. Summary and conclusions 380

46. Determination of All Solutions of a Decomposable Game 381

46.1. Elementary properties of decompositions 381

46.2. Decomposition and its relation to the solutions: First results con-
cerning F(e 0 ) 384

46.3. Continuation 386

46.4. Continuation 388

46.5. The complete result in F{e 0 ) 390

46.6. The complete result in E(e 0 ) 393

46.7. Graphical representation of a part of the result 394

46.8. Interpretation: The normal zone. Heredity of various properties 396

46.9. Dummies 397

46.10. Imbedding of a game 398

46.11. Significance of the normal zone 401

46.12. First occurrence of the phenomenon of transfer: n = 6 402

47. The Essential Three-Person Game in the New Theory 403

47.1. Need for this discussion 403

47.2. Preparatory considerations 403

47.3. The six cases of the discussion. Cases (I)— (III) 406

47.4. Case (IV) : First part 407

47.5. Case (IV) : Second part 409

47.6. Case (V) 413

47.7. Case (VI) 415

47.8. Interpretation of the result: The curves (one dimensional parts) in

the solution 416

47.9. Continuation: The areas (two dimensional parts) in the solution 418

CHAPTER X
SIMPLE GAMES

48. Winning and Losing Coalitions and Games Where They

Occur 420

48.1. The second type of 41.1. Decision by coalitions 420

48.2. Winning and Losing Coalitions 421

CONTENTS

49. Characterization of the Simple Games 423

49.1. General concepts of winning and losing coalitions 423

*49.2. The special role of one element sets 425

49.3. Characterization of the systems W, L of actual games 426

49.4. Exact definition of simplicity 428

49.5. Some elementary properties of simplicity 428

49.6. Simple games and their W, L. The Minimal winning coalitions: W m 429

49.7. The solutions of simple g^mes 430

50. The Majority Games and the Main Solution 431

50.1. Examples of simple games: The majority games 431

50.2. Homogeneity 433

50.3. A more direct use of the concept of imputation in forming solutions 435

50.4. Discussion of this direct approach 436

50.5. Connections with the general theory. Exact formulation 438

50.6. Reformulation of the result 440

50.7. Interpretation of the result 442

50.8. Connection with the Homogeneous Majority game. 443

51. Methods for the Enumeration of All Simple Games 445

51.1. Preliminary Remarks 445

51.2. The saturation method: Enumeration by means of W 446

51.3. Reasons for passing from W to W m . Difficulties of using W m 448

51.4. Changed Approach: Enumeration by means of W m 450

51.5. Simplicity and decomposition 452

51.6. Inessentiality, Simplicity and Composition. Treatment of the excess 454

51.7. A criterium of decomposability in terms of W m 455

52. The Simple Games for Small n 457

52.1. Program, n = 1, 2 play no role. Disposal of n — 3 457

52.2. Procedure for n ^4: The two element sets and their role in classify-
ing the W m 458

52.3. Decomposability of cases C *, C n - 2 , C n - 1 459

52.4. The simple games other than [1, • • • , 1, n — 2]* (with dummies):

The Cases C*, k = 0, 1, • • • , n — 3 461

52.5. Disposal of n — 4, 5 462

53. The New Possibilities of Simple Games for n ^ 6 463

53.1. The Regularities observed for n ^ 6 463

53.2. The six main counter examples (for n * 6, 7) 464

54. Determination of All Solutions in Suitable Games 470

54.1. Reasons to consider other solutions than the main solution in simple

games 470

54.2. Enumeration of those games for which all solutions are known 471

54.3. Reasons to consider the simple game [1, • ■ • , 1, » - 2]* 472

*55. The Simple Game [1, • • • , 1, n — 2]* 473

*55.1. Preliminary Remarks 473

*55.2. Domination. The chief player. Cases (I) and (II) 473

*55.3. Disposal of Case (I) 475

*55.4. Case (II): Determination of Y 478

*55.5. Case (II): Determination of Y 481

*55.6. Case (II): a and £* 484

CONTENTS

*55.7. Case (IT) and (II"). Disposal of Case (II') 485

*55.8^ Case (II"): a and V'. Domination 487

*55.9. Case (II"): Determination of V' 488

*55.10. Disposal of Case (II") 494

*55.11. Reformulation of the complete result 497

*55.12. Interpretation of the result 499

CHAPTER XI

GENERAL NON-ZERO-SUM GAMES

56. Extension of the Theory 504

56.1. Formulation of the problem 504

56.2. The fictitious player. The zero sum extension r 505

56.3. Questions concerning the character of r 506

56.4. Limitations of the use of r 508

56.5. The two possible procedures 510

56.6. The discriminatory solutions 511

56.7. Alternative possibilities 512

56.8. The new setup 514

56.9. Reconsideration of the case when T is a zero sum game 516

56.10. Analysis of the concept of domination 520

56.11. Rigorous discussion 523

56.12. The new definition of a solution 526

57. The Characteristic Function and Related Topics 527

57.1. The characteristic function: The extended and the restricted form 527

57.2. Fundamental properties 528

57.3. Determination of all characteristic functions 530

57.4. Removable sets of players 533

57.5. Strategic equivalence. Zero-sum and constant-sum games 535

58. Interpretation of the Characteristic Function 538

58.1. Analysis of the definition 538

58.2. The desire to make a gain vs. that to inflict a loss 539

58.3. Discussion 541

59. General Considerations 542

59.1. Discussion of the program 542

59.2. The reduced forms. The inequalities 543

59.3. Various topics 546

60. The Solutions of All General Games with n ^ 3 548

60.1. The case n = 1 548

60.2. The case n =» 2 549

60.3. The case n = 3 550

60.4. Comparison with the zero sum games 554

61. Economic Interpretation of the Results for n = 1, 2 555

61.1. The case n = 1 555

61.2. The case n =* 2. The two person market 555

61.3. Discussion of the two person market and its characteristic function 557

61.4. Justification of the standpoint of 58 559

61.5. Divisible goods. The “marginal pairs" 560

61.6. The price. Discussion 562

CONTENTS

62. Economic Interpretation of the Results for n = 3: Special

Case 564

62.1. The case n = 3, special case. The three person market 564

62.2. Preliminary discussion 566

62.3. The solutions: First subcase 566

62.4. The solutions: General form 569

62.5. Algebraical form of the result 570

62.6. Discussion 571

63. Economic Interpretation of the Results for n = 3 : General

Case 573

63.1. Divisible goods 573

63.2. Analysis of the inequalities 575

63.3. Preliminary discussion 577

63.4. The solutions 577

63.5. Algebraical form of the result 580

63.6. Discussion 581

64. The General Market 583

64.1. Formulation of the problem 583

64.2. Some special properties. Monopoly and monopsony 584

CHAPTER XII

EXTENSION OF THE CONCEPTS OF DOMINATION
AND SOLUTION

65. The Extension. Special Cases 587

65.1. Formulation of the problem 587

65.2. General remarks 588

65.3. Orderings, transitivity, acyclicity 589

65.4. The solutions: For a symmetric relation. For a complete ordering 591

65.5. The solutions: For a partial ordering 592

65.6. Acyclicity and strict acyclicity 594

65.7. The solutions: For an acyclic relation 597

65.8. Uniqueness' of solutions, acyclicity and strict acyclicity 600

65.9. Application to games: Discreteness and continuity 602

66. Generalization of the Concept of Utility 603

66.1. The generalization. The two phases of the theoretical treatment 603

66.2. Discussion of the first phase 604

66.3. Discussion of the second phase 606

66.4. Desirability of unifying the two phases 607

67. Discussion of an Example 608

67.1. Description of the example 608

67.2. The solution and its interpretation 611

67.3. Generalization: Different discrete utility scales 614

67.4. Conclusions concerning bargaining 616

Appendix: The Axiomatic Treatment of Utility 617

Index of Figures 633

Index of Names 634

Index of Subjects 635

CHAPTER I

FORMULATION OF THE ECONOMIC PROBLEM

1. The Mathematical Method in Economics

1.1. Introductory Remarks

1.1.1. The purpose of this book is to present a discussion of some funda-
mental questions of economic theory which require a treatment different
from that which they have found thus far in the literature. The analysis
is concerned with some basic problems arising from a study of economic
behavior which have been the center of attention of economists for a long
time. They have their origin in the attempts to find an exact description
of the endeavor of the individual to obtain a maximum of utility, or, in the
case of the entrepreneur, a maximum of profit. It is well known what
considerable — and in fact unsurmounted — difficulties this task involves
given even a limited number of typical situations, as, for example, in the
case of the exchange of goods, direct or indirect, between two or more
persons, of bilateral monopoly, of duopoly, of oligopoly, and of free compe-
tition. It will be made clear that the structure of these problems, familiar
to every student of economics, is in many respects quite different from the
way in which they are conceived at the present time. It will appear,
furthermore, that their exact positing and subsequent solution can only be
achieved with the aid of mathematical methods which diverge considerably
from the techniques applied by older or by contemporary mathematical
economists.

1 . 1 . 2 . Our considerations will lead to the application of the mathematical
theory of “games of strategy” developed by one of us in several successive
stages in 1928 and 1940-1941. 1 After the presentation of this theory, its
application to economic problems in the sense indicated above will be
undertaken. It will appear that it provides a new approach to a number of
economic questions as yet unsettled.

We shall first have to find in which way this theory of games can be
brought into relationship with economic theory, and what their common
elements are. This can be done best by stating briefly the nature of some
fundamental economic problems so that the common elements will be
seen clearly. It will then become apparent that there is not only nothing
artificial in establishing this relationship but that on the contrary this

1 The first phases of this- work were published: J. von Neumann , “Zur Theorie der
Gesellschaftsspiele,” Math. Annalen, vol. 100 (1928), pp. 295-320. The subsequent
completion of the theory, as well as the more detailed elaboration of the considerations
of loc. cit. above, are published here for the first time.

1

2 FORMULATION OF THE ECONOMIC PROBLEM

theory of games of strategy is the proper instrument with which to develop
a theory of economic behavior.

One would misunderstand the intent of our discussions by interpreting
them as merely pointing out an analogy between these two spheres. We
hope to establish satisfactorily, after developing a few plausible schematiza-
tions, that the typical problems of economic behavior become strictly
identical with the mathematical notions of suitable games of strategy.

1.2. Difficulties of the Application of the Mathematical Method

1.2.1. It may be opportune to begin with some remarks concerning the
nature of economic theory and to discuss briefly the question of the role
which mathematics may take in its development.

First let us be aware that there exists at present no universal system of
economic theory and that, if one should ever be developed, it will very
probably not be during our lifetime. The reason for this is simply that
economics is far too difficult a science to permit its construction rapidly,
especially in view of the very limited knowledge and imperfect description
of the facts with which economists are dealing. Only those wlio fail to
appreciate this condition are likely to attempt the construction of universal
systems. Even in sciences which are far more advanced than economics,
like physics, there is no universal system available at present.

To continue the simile with physics: It happens occasionally that a
particular physical theory appears to provide the basis for a universal
system, but in all instances up to the present time this appearance has not
lasted more than a decade at best. The everyday work of the research
physicist is certainly not involved with such high aims, but rather is con-
cerned with special problems which are “ mature/’ There would probably
be no progress at all in physics if a serious attempt were made to enforce
that super-standard. The physicist works on individual problems, some
of great practical significance, others of less. Unifications of fields which
were formerly divided and far apart may alternate with this type of work.
However, such fortunate occurrences are rare and happen only after each
field has been thoroughly explored. Considering the fact that economics
is much more difficult, much less understood, and undoubtedly in a much
earlier stage of its evolution as a science than physics, one should clearly not
expect more than a development of the above type in economics either.

Second we have to notice that the differences in scientific questions
make it necessary to employ varying methods which may afterwards have
to be discarded if better ones offer themselves. This has a double implica-
tion: In some branches of economics the most fruitful work may be that of
careful, patient description; indeed this may be by far the largest domain
for the present and for some time to come. In others it may be possible
to develop already a theory in a strict manner, and for that purpose the
use of mathematics may be required.

THE MATHEMATICAL METHOD IN ECONOMICS 3

Mathematics has actually been used in economic theory, perhaps even
in an exaggerated manner. In any case its use has not been highly suc-
cessful. This is contrary to what one observes in other sciences: There
mathematics has been applied with great success, and most sciences could
hardly get along without it. Yet the explanation for this phenomenon is
fairly simple.

1.2.2. It is not that there exists any fundamental reason why mathe-
matics should not be used in economics. The arguments often heard that
because of the human element, of the psychological factors etc., or because
there is — allegedly — no measurement of important factors, mathematics
will find no application, can all be dismissed as utterly mistaken. Almost
all these objections have been made, or might have been made, many
centuries ago in fields where mathematics is now the chief instrument of
analysis. This “ might have been” is meant in the following sense: Let
us try to imagine ourselves in the period which preceded the mathematical
or almost mathematical phase of the development in physics, that is the
16th century, or in chemistry and biology, that is the 18th century.
Taking for granted the skeptical attitude of those who object to mathe-
matical economics in principle, the outlook in the physical and biological
sciences at these early periods can hardly have been better than that in
economics — mutatis mutandis — at present.

As to the lack of measurement of the most important factors, the
example of the theory of heat is most instructive; before the development of
the mathematical theory the possibilities of quantitative measurements
were less favorable there than they are now in economics. The precise
measurements of the quantity and quality of heat (energy and temperature)
were the outcome and not the antecedents of the mathematical theory.
This ought to be contrasted with the fact that the quantitative and exact
notions of prices, money and the rate of interest were already developed
centuries ago.

A further group of objections against quantitative measurements in
economics, centers around the lack of indefinite divisibility of economic
quantities. This is supposedly incompatible with the use of the infini-
tesimal calculus and hence ( !) of mathematics. It is hard to see how such
objections can be maintained in view of the atomic theories in physics and
chemistry, the theory of quanta in electrodynamics, etc., and the notorious
and continued success of mathematical analysis within these disciplines.

At this point it is appropriate to mention another familiar argument of
economic literature which may be revived as an objection against the
mathematical procedure.

1.2.3. In order to elucidate the conceptions which we are applying to
economics, we have given and may give again some illustrations from
physics. There are many social scientists who object to the drawing of
such parallels on various grounds, among which is generally found the
assertion that economic theory cannot be modeled after physics since it is a

4 FORMULATION OF THE ECONOMIC PROBLEM

science of social, of human phenomena, has to take psychology into account,
etc. Such statements are at least premature. It is without doubt reason-
able to discover what has led to progress in other sciences, and to investigate
whether the application of the same principles may not lead to progress
in economics also. Should the need for the application of different principles
arise, it could be revealed only in the course of the actual development
of economic theory. This would itself constitute a major revolution.
But since most assuredly we have not yet reached such a state — and it is
by no means certain that there ever will be need for entirely different
scientific principles — it would be very unwise to consider anything else
than the pursuit of our problems in the manner which has resulted in the
establishment of physical science.

1 . 2 . 4 * The reason why mathematics has not been more successful in
economics must, consequently, be found elsewhere. The lack of real
success is largely due to a combination of unfavorable circumstances, some
of which can be removed gradually. To begin with, the economic problems
were not formulated clearly and are often stated in such vague terms as to
make mathematical treatment a priori appear hopeless because it is quite
uncertain what the problems really are. There is no point in using exact
methods where there i3 no clarity in the concepts and issues to which they
are to be applied. Consequently the initial task is to clarify the knowledge
of the matter by further careful descriptive work. But even in those
parts of economics where the descriptive problem has been handled more
satisfactorily, mathematical tools have seldom been used appropriately.
They were either inadequately handled, as in the attempts to determine a
general economic equilibrium by the mere counting of numbers of equations
and unknowns, or they led to mere translations from a literary form of
expression into symbols, without any subsequent mathematical analysis.

Next, the empirical background of economic science is definitely inade-
quate. Our knowledge of the relevant facts of economics is incomparably
smaller than that commanded in physics at the time when the mathe-
matization of that subject was achieved. Indeed, the decisive break which
came in physics in the seventeenth century, specifically in the field of
mechanics, was possible only because of previous developments in astron-
omy. It was backed by several millennia of systematic, scientific, astro-
nomical observation, culminating in an observer of unparalleled caliber,
Tycho de Brahe. Nothing of this sort has occurred in economic science. It
would have been absurd in physics to expect Kepler and Newton without
Tycho, — and there is no reason to hope for an easier development in
economics.

These obvious comments should not be construed, of course, as a
disparagement of statistical-economic research which holds the real promise
of progress in the proper direction.

It is due to the combination of the above mentioned circumstances
that mathematical economics has not achieved very much. The underlying

THE MATHEMATICAL METHOD IN ECONOMICS 5

vagueness and ignorance has not been dispelled by the inadequate and
inappropriate use of a powerful instrument that is very difficult to
handle.

In the light of these remarks we may describe our own position as follows:
The aim of this book lies not in the direction of empirical research. The
advancement of that side of economic science, on anything like the scale
which was recognized above as necessary, is clearly a task of vast propor-
tions. It may be hoped that as a result of the improvements of scientific
technique and of experience gained in other fields, the development of
descriptive economics will not take as much time as the comparison with
astronomy would suggest. But in any case the task seems to transcend
the limits of any individually planned program.

We shall attempt to utilize only some commonplace experience concern-
ing human behavior which lends itself to mathematical treatment and
which is of economic importance.

We believe that the possibility of a mathematical treatment of these
phenomena refutes the “fundamental” objections referred to in 1.2.2.

It will be seen, however, that this process of mathematization is not
at all obvious. Indeed, the objections mentioned above may have their
roots partly in the rather obvious difficulties of any direct mathematical
approach. We shall find it necessary to draw upon techniques of mathe-
matics which have not been used heretofore in mathematical economics, and
it is quite possible that further study may result in the future in the creation
of new mathematical disciplines.

To conclude, we may also observe that part of the feeling of dissatisfac-
tion with the mathematical treatment of economic theory derives largely
from the fact that frequently one is offered not proofs but mere assertions
which are really no better than the same assertions given in literary form.
Very frequently the proofs are lacking because a mathematical treatment
has been attempted of fields which are so vast and so complicated that for
a long time to come — until much more empirical knowledge is acquired —
there is hardly any reason at all to expect progress more mathematico.
The fact that these fields have been attacked in this way— as for example
the theory of economic fluctuations, the time structure of production, etc. —
indicates how much the attendant difficulties are being underestimated.
They are enormous and we are now in no way equipped for them.

1.2.5. We have referred to the nature and the possibilities of those
changes in mathematical technique — in fact, in mathematics itself — which
a successful application of mathematics to a new subject may produce.
It is important to visualize these in their proper perspective.

It must not be forgotten that these changes may be very considerable.
The decisive phase of the application of mathematics to physics — Newton's
creation of a rational discipline of mechanics — brought about, and can
hardly be separated from, the discovery of the infinitesimal calculus.
(There are several other examples, but none stronger than this.)

6 FORMULATION OF THE ECONOMIC PROBLEM

The importance of the social phenomena, the wealth and multiplicity
of theii manifestations, and the complexity of their structure, aie at least
equal to those in physics. It is therefore to be expected — or feared — that
mathematical discoveries of a stature comparable to that of calculus will
be needed in ordei to produce decisive success in this field. (Incidentally,
it is in this spirit that our present efforts must be discounted.) A fortiori
it is unlikely that a mere repetition of the tricks which served us so well in
physics will do for the social phenomena too. The probability is very slim
indeed, since it will be shown that we encounter in our discussions some
mathematical problems which are quite different from those which occur in
physical science.

These observations should be remembered in connection with the current
overemphasis on the use of calculus, differential equations, etc., as the
main tools of mathematical economics.

1.3. Necessary Limitations of the Objectives

1.3.1. We have to return, therefore, to the position indicated earlier:
It is necessary to begin with those problems which are described clearly,
even if they should not be as important from any other point of view. It
should be added, moreover, that a treatment of these manageable problems
may lead to results which are already fairly well known, but the exact
proofs may nevertheless be lacking. Before they have been given the
respective theory simply does not exist as a scientific theory. The move-
ments of the planets were known long before their courses had been calcu-
lated and explained by Newton's theory, and the same applies in many
smaller and less dramatic instances. And similarly in economic theory,
certain results — say the indeterminateness of bilateral monopoly — may be
known already. Yet it is of interest to derive them again from an exact
theory. The same could and should be said concerning practically all
established economic theorems.

1.3.2. It might be added finally that we do not propose to raise the
question of the practical significance of the problems treated. This falls
in line with what was said above about the selection of fields for theory.
The situation is not different here from that in other sciences. There too
the most important questions from a practical point of view may have been
completely out of reach during long and fruitful periods of their develop-
ment. This is certainly still the case in economics, where it is of utmost
importance to know how to stabilize employment, how to increase the
national income, or how to distribute it adequately. Nobody can really
answer these questions, and we need not concern ourselves with the pre-
tension that there can be scientific answers at present.

The great progress in every science came when, in the study of problems
which were modest as compared with ultimate aims, methods were devel-
oped which could be extended further and furthei. The free fall is a very
trivial physical phenomenon, but it was the study of this exceedingly simple

THE MATHEMATICAL METHOD IN ECONOMICS 7

fact and its comparison with the astronomical material, which brought forth
mechanics.

It seems to us that the same standard of modesty should be applied in
economics. It is futile to try to explain — and “ systematically ” at that —
everything economic. The sound procedure is to obtain first utmost
precision and mastery in a limited field, and then to proceed to another, some-
what wider one, and so on. This would also do away with the unhealthy
practice of applying so-called theories to economic or social reform where
they are in no way useful.

We believe that it is necessary to know as much as possible about the
behavior of the individual and about the simplest forms of exchange. This
standpoint was actually adopted with remarkable success by the founders
of the marginal utility school, but nevertheless it is not generally accepted.
Economists frequently point to much larger, more “burning” questions, and
brush everything aside which prevents them from making statements
about these. The experience of more advanced sciences, for example
physics, indicates that this impatience merely delays progress, including
that of the treatment of the “burning” questions. There is no reason to
assume the existence of shortcuts.

1.4. Concluding Remarks

1 . 4 . It is essential to realize that economists can expect no easier fate
than that which befell scientists in other disciplines. It seems reasonable
to expect that they will have to take up first problems contained in the very
simplest facts of economic life and try to establish theories which explain
them and which really conform to rigorous scientific standards. We can
have enough confidence that from then on the science of economics will
grow further, gradually comprising matters of more vital impoitance than
those with which one has to begin. 1

The field covered in this book is very limited, and we approach it in
this sense of modesty. We do not worry at all if the results of oui study
conform with views gained recently or held for a long time, for what is
important is the gradual development of a theory, based on a careful
analysis of the ordinary everyday interpretation of economic facts. This
preliminary stage is necessarily heuristic , i.e. the phase of transition from
unmathematical plausibility considerations to the formal procedure of
mathematics. The theory finally obtained must be mathematically rigor-
ous and conceptually general. Its first applications are necessarily to
elementary problems where the result has never been in doubt and no
theory is actually required. At this early stage the application selves to
corroborate the theory. The next stage develops when the theory is applied

1 The beginning is actually of a certain significance, because the forms of exchange
between a few individuals are the same as those observed on some of the most important
markets of modern industry, or in the case of barter exchange between states in inter-
national trade.

8 FORIVgJLATION OF THE ECONOMIC PROBLEM

to somewhat more complicated situations in which it may already lead to a
certain extent beyond the obvious and the familiar. Here theory and
application corroborate each other mutually. Beyond this lies the field of
real success: genuine prediction by theory. It is well known that all
mathematized sciences have gone through these successive phases of
evolution.

2. Qualitative Discussion of the Problem of Rational Behavior
2.1. The Problem of Rational Behavior

2.1.1. The subject matter of economic theory is the very complicated
mechanism of prices and production, and of the gaining and spending of
incomes. In the course of the development of economics it has been
found, and it is now well-nigh universally agreed, that an approach to this
vast problem is gained by the analysis of the behavior of the individuals
which constitute the economic community. This analysis has been pushed
fairly far in many respects, and while there still exists much disagreement
the significance of the approach cannot be doubted, no matter how great
its difficulties may be. The obstacles are indeed considerable, even if the
investigation should at first be limited to conditions of economics statics, as
they well must be. One of the chief difficulties lies in properly describing
the assumptions which have to be made about the motives of the individual.

* This problem has been stated traditionally by assuming that the consumer
desires to obtain a maximum of utility or satisfaction and the entrepreneur
a maximum of profits.

The conceptual and practical difficulties of the notion of utility, and
particularly of the attempts to describe it as a number, are well known and
their treatment is not among the primary objectives of this work. We shall
nevertheless be forced to discuss them in some instances, in particular in
3.3. and 3.5. Let it be said at once that the standpoint of the present book
on this very important and very interesting question will be mainly oppor-
tunistic. We wish to concentrate on one problem — which is not that of
the measurement of utilities and of preferences — and we shall therefore
attempt to simplify all other characteristics as far as reasonably possible.
We shall therefore assume that the aim of all participants in the economic
system, consumeis as well as entrepreneurs, is money, or equivalently a
single monetary commodity. This is supposed to be unrestrictedly divisible
and substitutable, freely transferable and identical, even in the quantitative
sense, with whatever “ satisfaction” or “ utility ” is desired by each par-
ticipant. (For the quantitative character of utility, cf. 3.3. quoted above.)

It is sometimes claimed in economic literature that discussions of the
notions of utility and preference are altogether unnecessary, since these are
purely verbal definitions with no empirically observable consequences, i.e.,
entirely tautological. It does not seem to us that these notions are quali-
tatively inferior to certain well established and indispensable notions in

THE PROBLEM OF RATIONAL BEHAVIOR

9

physics, like force, mass, charge, etc. That is, while they are in their
immediate form merely definitions, they become subject to empirical control
through the theories which are built upon them — and in no other way.
Thus the notion of utility is raised above the status of a tautology by such
economic theories as make use of it and the results of which can be compared
with experience or at least with common sense.

2.1.2. The individual who attempts to obtain these respective maxima
is also said to act “rationally.” But it may safely be stated that there
exists, at present, no satisfactory treatment of the question of rational
behavior. There may, for example, exist several ways by which to reach
the optimum position; they may depend upon the knowledge and under-
standing which the individual has and upon the paths of action open to
him. A study of all these questions in qualitative terms will not exhaust
them, because they imply, as must be evident, quantitative relationships.
It would, therefore, be necessary to formulate them in quantitative terms
so that all the elements of the qualitative description are taken into con-
sideration. This is an exceedingly difficult task, and we can safely say
that it has not been accomplished in the extensive literature about the
topic. The chief reason for this lies, no doubt, in the failure to develop
and apply suitable mathematical methods to the problem; this would
have revealed that the maximum problem which is supposed to correspond
to the notion of rationality is not at all formulated in an unambiguous way.
Indeed, a more exhaustive analysis (to be given in 4.3.-4.5.) reveals that
the significant relationships are much more complicated than the popular
and the “philosophical” use of the word “rational” indicates.

A valuable qualitative preliminary description of the behavior of the
individual is offered by the Austrian School, particularly in analyzing the
economy of the isolated “Robinson Crusoe.” We may have occasion to
note also some considerations of Bohm-Bawerk concerning the exchange
between two or more persons. The more recent exposition of the theory of
the individual’s choices in the form of indifference curve analysis builds up
on the very same facts or alleged facts but uses a method which is often held
to be superior in many ways. Concerning this we refer to the discussions in
2.1.1. and 3.3.

We hope, however, to obtain a real understanding of the problem of
exchange by studying it from an altogether different angle; this is, from the
perspective of a “game of strategy.” Our approach will become clear
presently, especially after some ideas which have been advanced, say by
Bohm-Bawerk — whose views may be considered only as a prototype of this
theory — are given correct quantitative formulation.

2.2. “Robinson Crusoe” Economy and Social Exchange Economy

2.2.1. Let us look more closely at the type of economy which is repre-
sented by the “Robinson Crusoe” model, that is an economy of an isolated
single person or otherwise organized under a single will. This economy is

10 FORMULATION OF THE ECONOMIC PROBLEM

confronted with certain quantities of commodities and a number of wants
which they may satisfy. The problem is to obtain a maximum satisfaction.
This is — considering in particular our above assumption of the numerical
character of utility — indeed an ordinary maximum problem, its difficulty
depending appaiently on the number of variables and on the nature of the
function to be maximized; but this is more of a practical difficulty than a
theoretical one. 1 If one abstracts from continuous production and from
the fact that consumption too stretches over time (and often uses durable
consumers 1 goods), one obtains the simplest possible model. It was
thought possible to use it as the very basis for economic theory, but this
attempt — notably a feature of the Austrian version — was often contested.
The chief objection against using this very simplified model of an isolated
individual for the theory of a social exchange economy is that it does not
represent an individual exposed to the manifold social influences. Hence,
it is said to analyze an individual who might behave quite differently if his
choices were made in a social world where he would be exposed to factors
of imitation, advertising, custom, and so on. These factors certainly make
a great difference, but it is to be questioned whether they change the formal
properties of the process of maximizing. Indeed the latter has never been
implied, and since we are concerned with this problem alone, we can leave
the above social considerations out of account.

Some other differences between “Crusoe” and a participant in a social
exchange economy will not concern us either. Such is the non-existence of
money as a means of exchange in the first case where there is only a standard
of calculation, for which purpose any commodity can serve. This difficulty
indeed has been ploughed under by our assuming in 2.1.2. a quantitative
and even monetary notion of utility. We emphasize again: Our interest
lies in the fact that even after all these drastic simplifications Crusoe is
confronted with a formal problem quite different from the one a participant
in a social economy faces.

2.2.2. Crusoe is given certain physical data (wants and commodities)
and his task is to combine and apply them in such a fashion as to obtain
a maximum resulting satisfaction. There can be no doubt that he controls
exclusively all the variables upon which this result depends — say the
allotting of resources, the determination of the uses of the same commodity
for different wants, etc. 2

Thus Crusoe faces an ordinary maximum problem, the difficulties of
which are of a purely technical — and not conceptual — nature, as pointed out.

2.2.3. Consider now a participant in a social exchange economy. His
problem has, of course, many elements in common with a maximum prob-

1 It is not important for the following to determine whether its theory is complete in
all its aspects.

* Sometimes uncontrollable factors also intervene, e.g. the weather in agriculture.
These however are purely statistical phenomena. Consequently they can be eliminated
by the known procedures of the calculus of probabilities: i.e., by determining the prob-
abilities of the various alternatives and by introduction of the notion of “mathematical
expectation.” Cf. however the influence on the notion of utility, discussed in 3.3.

THE PROBLEM OF RATIONAL BEHAVIOR 11

lem. But it also contains some, very essential, elements of an entirely
different nature. He too tries to obtain an optimum result. But in order
to achieve this, he must enter into relations of exchange with others. If
two or more persons exchange goods with each other, then the result for
each one will depend in general not merely upon his own actions but on
those of the others as well. Thus each participant attempts to maximize
a function (his above-mentioned “result”) of which he does not control all
variables. This is certainly no maximum problem, but a peculiar and dis-‘
concerting mixture of several conflicting maximum problems. Every parti-
cipant is guided by another principle and neither determines all variables
which affect his interest.

This kind of problem is nowhere dealt with in classical mathematics.
We emphasize at the risk of being pedantic that this is no conditional maxi-
mum problem, no problem of the calculus of variations, of functional
analysis, etc. It arises in full clarity, even in the most “elementary”
situations, e.g., when all variables can assume only a finite number of values.

A particularly sti iking expression of the popular misunderstanding
about this pseudo-maximum problem is the famous statement according to
which the purpose of social effort is the “greatest possible good for the
greatest possible number.” A guiding principle cannot be formulated
by the requirement of maximizing two (or more) functions at once.

■Such a principle, taken literally, is self-contradictory, (in general one
function will have no maximum where the other function has one.) It is
no better than saying, e.g., that a firm should obtain maximum prices
at maximum turnover, or a maximum revenue at minimum outlay. If
some order of importance of these principles or some weighted average is
meant, this should be stated. However, in the situation of the participants
in a social economy nothing of that sort is intended, but all maxima are
desired at once — by various participants.

One would be mistaken to believe that it can be obviated, like the
difficulty in the Crusoe case mentioned in footnote 2 on p. 10, by a mere
recourse to the devices of the theory of probability. Every participant can
determine the variables which describe his own actions but not those of the
others. Nevertheless those “alien ” variables cannot, from his point of view,
be described by statistical assumptions. This is because the others are
guided, just as he himself, by rational principles — whatever that may mean
— and no modus procedendi can be correct which does not attempt to under-
stand those principles and the interactions of the conflicting interests of all
participants.

Sometimes some of these interests run more oi less parallel — then we
are nearer to a simple maximum problem. But they can just as well be
opposed. The general theory must cover all these possibilities, all inter-
mediary stages, and all their combinations.

2.2.4. The difference between Crusoe’s perspective and that of a par-
ticipant in a social economy can also be illustrated in this way: Apart from

12 FORMULATION OF THE ECONOMIC PROBLEM

those variables which his will controls, Crusoe is given a number of data
which are “dead”; they are the unalterable physical background of the
situation. (Even when they are apparently variable, cf. footnote 2 on
p. 10, they are really governed by fixed statistical laws.) Not a single
datum with which he has to deal reflects another person’s will or intention
of an economic kind — based on motives of the same nature as his own. A
participant in a social exchange economy, on the other hand, faces data
of this last type as well : they are the product of other participants’ actions
and volitions (like prices). His actions will be influenced by his expectation
of these, and they in turn reflect the other participants’ expectation of his
actions.

Thus the study of the Crusoe economy and the use of the methods
applicable to it, is of much more limited value to economic theory than
has been assumed heretofore even by the most radical critics. The grounds
for this limitation lie not in the field of those social relationships which
we have mentioned before — although we do not question their significance —
but rather they arise from the conceptual differences between the original
(Crusoe’s) maximum problem and the more complex problem sketched above.

We hope that the reader will be convinced by the above that we face
here and now a really conceptual — and not merely technical — difficulty.
And it is this problem which the theory of “games of strategy” is mainly
devised to meet.

2.3. The Number of Variables and the Number of Participants

2.3.1. The formal set-up which we used in the preceding paragraphs to
indicate the events in a social exchange economy made use of a number of
“variables” which described the actions of the participants in this economy.
Thus every participant is allotted a set of variables, “his” variables, which
together completely describe his actions, i.e. express precisely the manifes-
tations of his will. We call these sets the partial sets of variables. The
partial sets of all participants constitute together the set of all variables, to
be called the total set. So the total number of variables is determined first
by the number of participants, i.e. of partial sets, and second by the number
of variables in every partial set.

From a purely mathematical point of view there would be nothing
objectionable in treating all the variables of any one partial set as a single
variable, “the” variable of the participant corresponding to this partial
set. Indeed, this is a procedure which we are going to use frequently in
our mathematical discussions; it makes absolutely no difference con-
ceptually, and it simplifies notations considerably.

For the moment, however, we propose to distinguish from each other the
variables within each partial set. The economic models to which one is
naturally led suggest that procedure; thus it is desirable to describe for
every participant the quantity of every particular good he wishes to acquire
by a separate variable, etc.

THE PROBLEM OF RATIONAL BEHAVIOR

13

2 . 3 . 2 . Now we must emphasize that any increase of the number of
variables inside a participant’s partial set may complicate our problem
technically, but only technically. Thus in a Crusoe economy — where
there exists only one participant and only one partial set which then coin-
cides with the total set — this may make the necessary determination of a
maximum technically more difficult, but it will not alter the “pure maxi-
mum” character of the problem. If, on the other hand, the number of
participants — i.e., of the partial sets of variables — is increased, something
of a very different nature happens. To use a terminology which will turn
out to be significant, that of games, this amounts to an increase in the
number of players in the game. However, to take the simplest cases, a
three-person game is very fundamentally different from a two-person game,
a four-person game from a three-person game, etc. The combinatorial
complications of the problem — which is, as we saw, no maximum problem
at all — increase tremendously with every increase in the number of players,
— as our subsequent discussions will amply show.

We have gone into this matter in such detail particularly because in
most models of economics a peculiar mixture of these two phenomena occurs.
Whenever the number of players, i.e. of participants in a social economy,
increases, the complexity of the economic system usually increases too;
e.g. the number of commodities and services exchanged, processes of
production used, etc. Thus the number of variables in every participant’s
partial set is likely to increase. But the number of participants, i.e. of
partial sets, has increased too. Thus both of the sources which we discussed
contribute pari passu to the total increase in the number of variables. It is
essential to visualize each source in its proper role.

2.4. The Case of Many Participants : Free Competition

2 . 4 . 1 . In elaborating the contrast between a Crusoe economy and a
social exchange economy in 2.2.2.-2.2.4., we emphasized those features
of the latter which become more prominent when the number of participants
— while greater than 1 — is of moderate size. The fact that every partici-
pant is influenced by the anticipated reactions of the others to his own
measures, and that this is true for each of the participants, is most strikingly
the crux of the matter (as far as the sellers are concerned) in the classical
problems of duopoly, oligopoly, etc. When the number of participants
becomes really great, some hope emerges that the influence of every par-
ticular participant will become negligible, and that the above difficulties
may recede and a more conventional theory become possible. These
are, of course, the classical conditions of “free competition.” Indeed, this
was the starting point of much of what is best in economic theory. Com-
pared with this case of great numbers — free competition — the cases of small
numbers on the side of the sellers — monopoly, duopoly, oligopoly — were
even considered to be exceptions and abnormities. (Even in these cases
the number of participants is still very large in view of the competition

14 FORMULATION OF THE ECONOMIC PROBLEM

among the buyers. The cases involving really small numbers are those of
bilateral monopoly, of exchange between a monopoly and an oligopoly, or
two oligopolies, etc.)

2.4.2. In all fairness to the traditional point of view this much ought
to be said: It is a well known phenomenon in many branches of the exact
and physical sciences that very great numbers are often easier to handle
than those of medium size. An almost exact theory of a gas, containing
about 10 26 freely moving particles, is incomparably easier than that of the
solar system, made up of 9 major bodies; and still more than that of a mul-
tiple star of three or four objects of about the same size. This is, of course,
due to the excellent possibility of applying the laws of statistics and prob-
abilities in the first case.

This analogy, however, is far from perfect for our problem. The theory
of mechanics for 2, 3, 4, * • • bodies is well known, and in its general
theoretical (as distinguished from its special and computational) form is the
foundation of the statistical theory for great numbers. For the social
exchange economy — i.e. for the equivalent “ games of strategy ” — the theory
of 2, 3, 4, • • participants was heretofore lacking. It is this need that
our previous discussions were designed to establish and that our subsequent
investigations will endeavor to satisfy. In other words, only after the
theory for moderate numbers of participants has been satisfactorily devel-
oped will it be possible to decide whether extremely great numbers of par-
ticipants simplify the situation. Let us say it again: We share the hope —
chiefly because of the above-mentioned analogy in other fields! — that such
simplifications will indeed occur. The current assertions concerning free
competition appear to be very valuable surmises and inspiring anticipations
of results. But they are not results and it is scientifically unsound to treat
them as such as long as the conditions which we mentioned above are not
satisfied.

There exists in the literature a considerable amount of theoretical dis-
cussion purporting to show that the zones of indeterminateness (of rates of
exchange) — which undoubtedly exist when the number of participants is
small — narrow and disappear as the number increases. This then would
provide a continuous transition into the ideal case of free competition — for
a very great number of participants — where all solutions would be sharply
and uniquely determined. While it is to be hoped that this indeed turns out
to be the case in sufficient generality, one cannot concede that anything
like this contention has been established conclusively thus far. There is
no getting away from it: The problem must be formulated, solved and
understood for small numbers of participants before anything can be proved
about the changes of its character in any limiting case of large numbers,
such as free competition.

2.4.3. A really fundamental reopening of this subject is the more
desirable because it is neither certain nor probable that a mere increase in
the number of participants will always lead in fine to the conditions of

THE NOTION OF UTILITY

15

free competition. The classical definitions of free competition all involve
further postulates besides the greatness of that number. E.g., it is clear
that if certain great groups of participants will — for any reason whatsoever —
act together, then the great number of participants may not become
effective; the decisive exchanges may take place directly between large
“coalitions,” 1 few in number, and not between individuals, many in number,
acting independently. Our subsequent discussion of “games of strategy”
will show that the role and size of “coalitions” is decisive throughout the
entire subject. Consequently the above difficulty — though not new — still
remains the crucial problem. Any satisfactory theory of the “limiting
transition” from small numbers of participants to large numbers will have
to explain under what circumstances such big coalitions will or will not be
formed — i.e. when the large numbers of participants will become effective
and lead to a more or less free competition. Which of these alternatives is
likely to arise will depend on the physical data of the situation. Answering
this question is, we think, the real challenge to any theory of free competition.

2.5. The “Lausanne” Theory

2.6. This section should not be concluded without a reference to the
equilibrium theory of the Lausanne School and also of various other systems
which take into consideration “individual planning” and interlocking
individual plans. All these systems pay attention to the interdependence
of the participants in a social economy. This, however, is invariably done
under far-reaching restrictions. Sometimes free competition is assumed,
after the introduction of which the participants face fixed conditions and
act like a number of Robinson Crusoes — solely bent on maximizing their
individual satisfactions, which under these conditions are again independent.
In other cases other restricting devices are used, all of which amount to
excluding the free play of “coalitions” formed by any or all types of par-
ticipants. There are frequently definite, but sometimes hidden, assump-
tions concerning the ways in which their partly parallel and partly opposite
interests will influence the participants, and cause them to cooperate or not,
as the case may be. We hope we have shown that such a procedure amounts
to a petitio principii — at least on the plane on which we should like to put
the discussion. It avoids the real difficulty and deals with a verbal problem,
which is not the empirically given one. Of course we do not wish to ques-
tion the significance of these investigations — but they do not answer our
queries.

3. The Notion of Utility

3.1. Preferences and Utilities

3.1.1. We have stated already in 2.1.1. in what way w r e wish to describe
the fundamental concept of individual preferences by the use of a rather

1 Such as trade unions, consumers' cooperatives, industrial cartels, and conceivably
some organizations more in the political sphere.

16

FORMULATION OF THE ECONOMIC PROBLEM

far-reaching notion of utility. Many economists will feel that we are
assuming far too much (cf . the enumeration of the properties we postulated
in 2.1.1.), and that our standpoint is a retrogression from the more cautious
modern technique of “ indifference curves.”

Before attempting any specific discussion let us state as a general
excuse that our procedure at worst is only the application of a classical
preliminary device of scientific analysis: To divide the difficulties, i.e. to
concentrate on one (the subject proper of the investigation in hand), and
to reduce all others as far as reasonably possible, by simplifying and schema-
tizing assumptions. We should also add that this high handed treatment
of preferences and utilities is employed in the main body of our discussion,
but we shall incidentally investigate to a certain extent the changes which an
avoidance of the assumptions in question would cause in our theory (cf. 66.,
67.).

We feel, however, that one part of our assumptions at least — that of
treating utilities as numerically measurable quantities — is not quite as
radical as is often assumed in the literature. We shall attempt to prove
this particular point in the paragraphs which follow. It is hoped that the
reader will forgive us for discussing only incidentally in a condensed form
a subject of so great a conceptual importance as that of utility. It seems
however that even a few remarks may be helpful, because the question
of the measurability of utilities is similar in character to corresponding
questions in the physical sciences.

3.1.2. Historically, utility was fiist conceived as quantitatively measur-
able, i.e. as a number. Valid objections can be and have been made against
this view in its original, naive form. It is clear that every measurement —
or rather every claim of measurability — must ultimately be based on some
immediate sensation, which possibly cannot and certainly need not be
analyzed any further. 1 In the case of utility the immediate sensation of
preference — of one object or aggregate of objects as against another —
provides this basis. But this permits us only to say when for one person
one utility is greater than another. It is not in itself a basis for numerical
comparison of utilities for one person nor of any comparison between
different persons. Since there is no intuitively significant way to add two
lUtilities for the same person, the assumption that utilities are of non-
pumerical character even seems plausible. The modern method of indiffer-
ence curve analysis is a mathematical procedure to describe this situation.

3.2. Principles of Measurement : Preliminaries

3.2.1. All this is strongly reminiscent of the conditions existant at the
beginning of the theory of heat: that too was based on the intuitively clear
concept of one body feeling warmer than another, yet there was no immedi-
ate way to express significantly by how much, or how many times, or in
what sense.

1 Such as the sensations of light, heat, muscular effort, etc., in the corresponding
branches of physics.

THE NOTION OF UTILITY

17

This comparison with heat also shows how little one can forecast a priori
what the ultimate shape of such a theory will be. The above crude indica-
tions do not disclose at all what, as we now know, subsequently happened.
It turned out that heat permits quantitative description not by one numbei
but by two: the quantity of heat and temperature. The former is rather
directly numerical because it turned out to be additive and also in an
unexpected way connected with mechanical energy which was numerical
anyhow. The latter is also numerical, but in a much more subtle way;
it is not additive in any immediate sense, but a rigid numerical scale for it
emerged from the study of the concordant behavior of ideal gases, and the
role of absolute temperature in connection with the entropy theorem.

3.2.2. The historical development of the theory of heat indicates that
one must be extremely careful in making negative assertions about any
concept with the claim to finality. Even if utilities look very unnumerical
today, the history of the experience in the theory of heat may repeat itself,
and nobody can foretell with what ramifications and variations. 1 And it
should certainly not discourage theoretical explanations of the formal
possibilities of a numerical utility.

3.3. Probability and Numerical Utilities

3.3.1. We can go even one step beyond the above double negations —
which were only cautions against premature assertions of the impossibility
of a numerical utility. It can be shown that under the conditions on which
the indifference curve analysis is based very little extra effort is needed to
reach a numerical utility.

It has been pointed out repeatedly that a numerical utility is dependent
upon the possibility of comparing differences in utilities. This may seem —
and indeed is — a more far-reaching assumption than that of a mere ability
to state preferences. But it will seem that the alternatives to which eco-
nomic preferences must be applied are such as to obliterate this distinction.

3.3.2. ILet us for the moment accept the picture of an individual whose
system of preferences is all-embracing ^nd^eomplete, Le^who^for any two
objects or rather for any two imagined events, possesses a clear intuition of
preference.

More precisely we expect him, for any two alternative events which are
put before him as possibilities, to be able to tell which of the two he prefers.

It is a very natural extension of this picture to permit such an individual
to compare not only events, but even combinations of events with stated
probabilities. 2

By a combination of two events we mean this: Let the two events be
denoted by B and C and use, for the sake of simplicity, the probability

1 A good example of the wide variety of formal possibilities is given by the entirely
different development of the theory of light, colors, and wave lengths. All these notions
too became numerical, but in an entirely different way.

* Indeed this is necessary if he is engaged in economic activities which are explicitly
dependent on probability. Cf . the example of agriculture in footnote 2 on p. 10.

18

FORMULATION OF THE ECONOMIC PROBLEM

50%-50%. Then the “combination” is the prospect of seeing B occur
with a probability of 50% and (if B does not occur) C with the (remaining)
probability of 50%. We stress that the two alternatives are mutually
exclusive, so that no possibility of complementarity and the like exists.
Also, that an absolute certainty of the occurrence of either B or C exists.

To restate our position. We expect the individual under consideration
to possess a clear intuition whether he prefers the event A to the 50-50
combination of B or C, or conversely. It is clear that if he prefers A to B
and also to C, then he will prefer it to the above combination as well;
similarly, if he prefers B as well as C to A, then he will prefei the combination
too. But if he should prefer A to, say B , but at the same time C to A, then
any assertion about his preference of A against the combination contains
fundamentally new information. Specifically: If he now prefers A to the
50-50 combination of B and C, this provides a plausible base for the numer-
ical estimate that his preference of A over B is in excess of his preference of
C over A. 1 * 2 * * * * *

If this standpoint is accepted, then there is a criterion with which to
compare the preference of C over A with the preference of A over B. It is
well known that thereby utilities — or rather differences of utilities — become
numerically measurable.

That the possibility of comparison between A, B , and C only to this
extent is already sufficient for a numerical measurement of “distances”
was first observed in economics by Pareto. Exactly the same argument
has been made, however, by Euclid for the position of points on a line — in
fact it is the very basis of his classical derivation of numerical distances.

The introduction of numerical measures can be achieved even more
directly if use is made of all possible probabilities. Indeed: Consider
three events, C, A, B, for which the order of the individual's preferences
is the one stated. Let a be a real number between 0 and 1, such that A
is exactly equally desirable with the combined event consisting of a chance
of probability 1 — a for B and the remaining chance of probability a for C.
Then we suggest the use of a as a numerical estimate for the ratio of the
preference of A over B to that of C over B. 8 An exact and exhaustive

1 To give a simple example: Assume that an individual prefers the consumption of a
glass of tea to that of a cup of coffee, and the cup of coffee to a glass of milk. If we now
want to know whether the last preference — i.e., difference in utilities — exceeds the former,
it suffices to place him in a situation where he must decide this: Does he prefer a cup of

coffee to a glass the content of which will be determined by a 50 %-50 % chance device as

tea or milk.

* Observe that we have only postulated an individual intuition which permits decision

as to which of two “ events ” is preferable. But we have not directly postulated any
intuitive estimate of the relative sizes of two preferences — i.e. in the subsequent termi-
nology, of two differences of utilities.

This is important, since the former information ought to be obtainable in a reproduci-
ble way by mere “questioning.”

1 Ibis offers a good opportunity for another illustrative example. The above tech-
nique permits a direct determination of the ratio q of the utility of possessing 1 unit of a

certain good to the utility of possessing 2 units of the same good. The individual must

THE NOTION OF UTILITY

19

elaboration of these ideas requires the use of the axiomatic method. A sim-
ple treatment on this basis is indeed possible. We shall discuss it in
3.5-3.7.

3 . 3 . 3 . To avoid misunderstandings let us state that the “events”
which were used above as the substratum of preferences are conceived as
future events so as to make all logically possible alternatives equally
admissible. However, it would be an unnecessary complication, as far
as our present objectives are concerned, to get entangled with the problems
of the preferences between events in different periods of the future. 1 It
seems, however, that such difficulties can be obviated by locating all
“events” in which we are interested at one and the same, standardized,
moment, preferably in the immediate future.

The above considerations are so vitally dependent upon the numerical
concept of probability that a few words concerning the latter may be
appropriate.

Probability has often been visualized as a subjective concept more
or less in the nature of an estimation. Since we propose to use it in con-
structing an individual, numerical estimation of utility, the above view of
probability would not serve our purpose. The simplest procedure is, there-
fore, to insist upon the alternative, perfectly well founded interpretation of
probability as frequency in long runs. This gives directly the necessary
numerical foothold. 2

3 . 3 . 4 . This procedure for a numerical measurement of the utilities of the
individual depends, of course, upon the hypothesis of completeness in the
system of individual preferences. 8 It is conceivable — and may even in a
way be more realistic — to allow for cases where the individual is neither
able to state which of two alternatives he prefers nor that they are equally
desirable. In this case the treatment by indifference curves becomes
impracticable too. 4

How real this possibility is, both for individuals and for organizations,
seems to be an extremely interesting question, but it is a question of fact.
It certainly deserves further study. We shall reconsider it briefly in 3.7.2.

At any rate we hope we have shown that the treatment by indifference
curves implies either too much or too little: if the preferences of the indi-

be given the choice of obtaining 1 unit with certainty or of playing the chance to get two
units with the probability «, or nothing with the probability 1 — a. If he prefers the
former, then a < q; if he prefers the latter, then a > q; if he cannot state a preference
either way, then a = q.

1 It is well known that this presents very interesting, but as yet extremely obscure,
connections with the theory of saving and interest, etc.

* If one objects to the frequency interpretation of probability then the two concepts
(probability and preference) can be axiomatized together. This too leads to a satis-
factory numerical concept of utility which will be discussed on another occasion.

1 We have not obtained any basis for a comparison, quantitatively or qualitatively,
of the utilities of different individuals.

4 These problems belong systematically in the mathematical theory of ordered sets.
The above question in particular amounts to asking whether events, with respect to
preference, form a completely or a partially ordered set. Cf. 65.3.

20 FORMULATION OF THE ECONOMIC PROBLEM

vidual are not all comparable, then the indifference curves do not exist. 1
If the individual's preferences are all comparable, then we can even obtain a
(uniquely defined) numerical utility which renders the indifference curves
superfluous.

All this becomes, of course, pointless for the entrepreneur who can
calculate in terms of (monetary) costs and profits.

3 . 3 . 5 . The objection could be raised that it is not necessary to go into
all these intricate details concerning the measurability of utility, since
evidently the common individual, whose behavior one wants to describe,
does not measure his utilities exactly but rather conducts his economic
activities in a sphere of considerable haziness. The same is true, of course,
for much of his conduct regarding light, heat, muscular effort, etc. But in
order to build a science of physics these phenomena had to be measured.
And subsequently the individual has come to use the results of such measure-
ments — directly or indirectly — even in his everyday life. The same may
obtain in economics at a future date. Once a fuller understanding of
economic behavior has been achieved with the aid of a theory which makes
use of this instrument, the life of the individual might be materially affected.
It is, therefore, not an unnecessary digression to study these problems.

3.4. Principles of Measurement : Detailed Discussion

3 . 4 . 1 . The reader may feel, on the basis of the foregoing, that we
obtained a numerical scale of utility only by begging the principle, i.e. by
really postulating the existence of such a scale. We have argued in 3.3.2.
that if an individual prefers A to the 50-50 combination of B and C (while
preferring C to A and A to B), this provides a plausible basis for the numer-
ical estimate that this preference of A over B exceeds that of C over A .
Are we not postulating here — or taking it for granted — that one preference
may exceed another, i.e. that such statements convey a meaning? Such
a view would be a complete misunderstanding of our procedure.

3 . 4 . 2 . We are not postulating — or assuming — anything of the kind. We
have assumed only one thing — and for this there is good empirical evidence
— namely that imagined events can be combined with probabilities. And
therefore the same must be assumed for the utilities attached to them, —
whatever they may be. Or to put it in more mathematical language:

There frequently appear in science quantities which are a priori not
mathematical, but attached to certain aspects of the physical world.
Occasionally these quantities can be grouped together in domains within
which certain natural, physically defined operations are possible. Thus
the physically defined quantity of “mass” permits the operation of addition.
The physico-geometrically defined quantity of “distance” 2 permits the same

1 Points on the same indifference curve must be identified and are therefore no
instances of incomparability.

* Let us, for the sake of the argument, view geometry as a physical discipline, — a
sufficiently tenable viewpoint. By “geometry” we mean — equally for the sake of the
argument — Euclidean geometry.

THE NOTION OF UTILITY

21

operation. On the other hand, the physico-geometrically defined quantity
of “ position” does not permit this operation , 1 but it permits the operation
of forming the “ center of gravity” of two positions . 2 Again other physico-
geometrical concepts, usually styled “ vectorial” — like velocity and accelera-
tion — permit the operation of “ addition.”

3.4.3. In all these cases where such a “ natural” operation is given a
name which is reminiscent of a mathematical operation — like the instances
of “addition” above — one must carefully avoid misunderstandings. This
nomenclature is not intended as a claim that the two operations with the
same name are identical, — this is manifestly not the case; it only expresses
the opinion that they possess similar traits, and the hope that some cor-
respondence between them will ultimately be established. This of course —
when feasible at all — is done by finding a mathematical model for the
physical domain in question, within which those quantities are defined by
numbers, so that in the model the mathematical operation describes the
synonymous “natural” operation.

To return to our examples: “energy” and “mass” became numbers in
the pertinent mathematical models, “natural” addition becoming ordinary
addition. “Position” as well as the vectorial quantities became triplets 3 4 5 of
numbers, called coordinates or components respectively. The “natural”
concept of “center of gravity” of two positions {xi, x 2 , x 3 } and \x\, x 2 , x\\*
with the “masses” a, 1 — a (cf. footnote 2 above), becomes

{ax 1 + (1 - a)x[, ax 2 + (1 — a)xj, ax 3 + (1 — a)x 2 \}

The “natural” operation of “addition” of vectors {x h x 2 , x z \ and [x[, x 2 , x 2 \
becomes [x\ + x [ , x 2 + x 2 , x z + £ 3 } . 6

What was said above about “natural” and mathematical operations
applies equally to natural and mathematical relations. The various con-
cepts of “greater” which occur in physics — greater energy, force, heat,
velocity, etc. — are good examples.

These “natural” relations are the best base upon which to construct
mathematical models and to correlate the physical domain with them . 7 ' 8

1 We are thinking of a “homogeneous” Euclidean space, in which no origin or frame of
reference is preferred above any other.

2 With respect to two given masses a, 0 occupying those positions. It may be con-
venient to normalize so that the total mass is the unit, i.e. 0 = 1 — a.

3 We are thinking of three-dimensional Euclidean space.

4 We are now describing them by their three numerical coordinates.

5 This is usually denoted by a (xi, Xi, 1 + (1 — a) \x[, x' tl x\\. Cf. (16:A:c) in 16.2.1.

6 This is usually denoted by { xi, x *, x 8 ) -f (xj, x 2j x\ | . Cf. the beginning of 16.2.1.

7 Not the only one. Temperature is a good counter-example. The “natural” rela-
tion of “greater” would not have sufficed to establish the present day mathematical
model, — i.e. the absolute temperature scale. The devices actually used were different.

cf. 3 . 2 . 1 .

8 We do not want to give the misleading impression of attempting here a complete
picture of the formation of mathematical models, i.e. of physical theories. It should be
remembered that this is a very varied process with many unexpected phases. An impor-
tant one is, e.g., the disentanglement of concepts: i.e. splitting up something which at

22

FORMULATION OF THE ECONOMIC PROBLEM

3.4.4. Here a further remark must be made. Assume that a satisfactory
mathematical model for a physical domain in the above sense has been
found, and that the physical quantities under consideration have been
correlated with numbers. In this case it is not true necessarily that the
description (of the mathematical model) provides for a unique way of
correlating the physical quantities to numbers; i.e., it may specify an entire
family of such correlations — the mathematical name is mappings — any
one of which can be used for the purposes of the theory. Passage from one
of these correlations to another amounts to a transformation of the numerical
data describing the physical quantities. We then say that in this theory
the physical quantities in question are described by numbers up to that
system of transformations. The mathematical name of such transformation
systems is groups 4

Examples of such situations are numerous. Thus the geometrical con-
cept of distance is a number, up to multiplication by (positive) constant
factors. 2 The situation concerning the physical quantity of mass is the
same. The physical concept of energy is a number up to any linear trans-
formation, — i.e. addition of any constant and multiplication by any (posi-
tive) constant. 3 The concept of position is defined up to an inhomogeneous
orthogonal linear transformation. 4 6 The vectorial concepts are defined
up to homogeneous tiansformations of the same kind. 6 6

3.4.6. It is even conceivable that a physical quantity is a number up to
any monotone transformation. This is the case for quantities for which
only a “natural” relation “greater” exists — and nothing else. E.g. this
was the case for temperature as long as only the concept of “warmer" was
known; 7 it applies to the Mohs' scale of hardness of minerals; it applies to

superficial inspection seems to be one physical entity into several mathematical notions.
Thus the “ disentanglement ” of force and energy, of quantity of heat and temperature,
were decisive in their respective fields.

It is quite unforeseeable how many such differentiations still lie ahead in economic
theory.

1 We shall encounter groups in another context in 28.1.1, where references to the
literature are also found.

* I.e. there is nothing in Euclidean geometry to fix a unit of distance.

* I.e. there is nothing in mechanics to fix a zero or a unit of energy. Cf. with footnote 2
above. Distance has a natural zero, — the distance of any point from itself.

4 I.e. l*i, x*, *i | are to be replaced by (*i*, x 2 *, x%*\ where

*i* « flnii 4- a u x t 4* ai8*j 4 hi,

Xt * — a* 1*1 4 022*1 4 028*i 4 h 2 ,

^1* * 081*1 4 082*2 4 088*8 4 h|,

the a, 7 , h»* being constants, and the matrix ( 0 ,,) what is known as orthogonal.

4 I.e. there is nothing in geometry to fix either origin or the frame of reference when
positions are concerned; and nothing to fix the frame of reference when vectors are
concerned.

* I.e. the bi =■ 0 in footnote 4 above. Sometimes a wider concept of matrices is
permissible, — all those with determinants ^ 0. We need not discuss these matters here.

* But no quantitatively reproducible method of thermometry.

THE NOTION OF UTILITY

23

the notion of utility when this is based on the conventional idea of prefer-
ence. In these cases one may be tempted to take the view that the quantity
in question is not numerical at all, considering how arbitrary the description
by numbers is. It seems to be preferable, however, to refrain from such
qualitative statements and to state instead objectively up to what system
of transformations the numerical description is determined. The case
when the system consists of all monotone transformations is, of course, a
rather extreme one; various graduations at the other end of the scale are
the transformation systems mentioned above: inhomogeneous or homo-
geneous orthogonal linear transformations in space, linear transformations
of one numerical variable, multiplication of that variable by a constant. 1
In fine , the case even occurs where the numerical description is absolutely
rigorous, i.e. where no transformations at all need be tolerated. 2

3.4.6. Given a physical quantity, the system of transformations up to
which it is described by numbers may vary in time, i.e. with the stage of
development of the subject. Thus temperature was originally a number
only up to any monotone transformation. 8 With the development of
thermometry — particularly of the concordant ideal gas thermometry — the
transformations were restricted to the linear ones, i.e. only the absolute
zero and the absolute unit were missing. Subsequent developments of
thermodynamics even fixed the absolute zero so that the transformation
system in thermodynamics consists only of the multiplication by constants.
Examples could be multiplied but there seems to be no need to go into this
subject further.

For utility the situation seems to be of a similar nature. One may
take the attitude that the only “ natural ’ 1 datum in this domain is the
relation “greater,” i.e. the concept of preference. In this case utilities are
numerical up to a monotone transformation. This is, indeed, the generally
accepted standpoint in economic literature, best expressed in the technique
of indifference curves.

To narrow the system of transformations it would be necessary to dis-
cover further “ natural” operations or relations in the domain of utility.
Thus it was pointed out by Pareto 4 that an equality relation for utility
differences would suffice; in our terminology it would reduce the transfor-
mation system to the linear transformations. 5 However, since it does not

1 One could also imagine intermediate cases of greater transformation systems than
these but not containing all monotone transformations. Various forms of the theory of
relativity give rather technical examples of this.

* In the usual language this would hold for physical quantities where an absolute zero
as well as an absolute unit can be defined. This is, e.g., the case for the absolute value
(not the vector!) of velocity in such physical theories as those in which light velocity
plays a normative role: Maxwellian electrodynamics, special relativity.

3 As long as only the concept of “warmer” — i.e. a “natural” relation “greater” — was
known. We discussed this in extenao previously.

4 V. Pareto, Manuel d’Economie Politique, Paris, 1907, p. 264.

‘This is exactly what Euclid did for position on a line. The utility concept of
“preference” corresponds to the relation of “lying to the right of” there, and the (desired)
relation of the equality of utility differences to the geometrical congruence of intervals.

24

FORMULATION OF THE ECONOMIC PROBLEM

seem that this relation is really a “natural” one — i.e. one which can be
interpreted by reproducible observations — the suggestion does not achieve
the purpose.

3.5. Conceptual Structure of the Axiomatic Treatment of Numerical Utilities

3.6.1. The failure of one particular device need not exclude the possibility
of achieving the same end by another device. Our contention is that the
domain of utility contains a “natural” operation which narrows the system
of transformations to precisely the same extent as the other device would
have done. This is the combination of two utilities with two given alterna-
tive probabilities a, 1 — a, (0 < a < 1) as described in 3.3.2. The
process is so similar to the formation of centers of gravity mentioned in
3.4.3. that it may be advantageous to use the same terminology. Thus
we have for utilities u , v the “natural” relation u > v (read: u is preferable
to v ), and the “natural” operation au + ( 1 — a)v, (0 < a < 1), (read:
center of gravity of u ) v with the respective weights a, 1 — a; or: combina-
tion of u, v with the alternative probabilities a, l — a). If the existence —
and reproducible observability — of these concepts is conceded, then our
way is clear: We must find a correspondence between utilities and numbers
which carries the relation u > v and the operation au + (1 — a)v for
utilities into the synonymous concepts for numbers.

Denote the correspondence by

u — » p = v(u),

u being the utility and \(u) the number which the correspondence attaches
to it. Our requirements are then:

(3:l:a) u > v implies v(u) > v(v),

(3:1 :b) v(au + (1 — a)v) = av(u) + (1 — a)v(v). 1

If two such correspondences

(3:2:a) u-+p = v(u),

(3:2:b) u -> p' = v'(u),

should exist, then they set up a correspondence between numbers

(3:3) p^p',

for which we may also write

(3:4) P ' = *( P ).

Since (3:2:a), (3:2:b) fulfill (3:1 :a), (3:1 :b), the correspondence (3:3), i.e.
the function 0(p) in (3:4) must leave the relation p > <r 2 and the operation

1 Observe that in in each case the left-hand side has the “natural” concepts for
utilities, and the right-hand side the conventional ones for numbers.

* Now these are applied to numbers p, <r!

THE NOTION OF UTILITY

25

ap + (1 — a)a unaffected (cf footnote 1 on p. 24). I.e.

(3:5:a) p > a implies <£(p) > <£(<r),

(3:5:b) <f>(ap + (1 — a)a) = a<f>(p) + (1 — a)4>(<r).

Hence </>(p) must be a linear function, i.e.

(3:6) p' = <f>(p) s co 0 p + «i,

where coo, coi are fixed numbers (constants) with w 0 > 0.

So we see: If such a numerical valuation of utilities 1 exists at all, then
it is determined up to a linear transformation. 2 * 8 I.e. then utility is a
number up to a linear transformation.

In order that a numerical valuation in the above sense should exist it
is necessary to postulate certain properties of the relation u > v and the
operation au + (1 — a)v for utilities. The selection of these postulates
or axioms and their subsequent analysis leads to problems of a certain
mathematical interest. In what follows we give a general outline of the
situation for the orientation of the reader; a complete discussion is found in
the Appendix.

3 . 6 . 2 . A choice of axioms is not a purely objective task. It is usually
expected to achieve some definite aim — some specific theorem or theorems
are to be derivable from the axioms — and to this extent the problem is
exact and objective. But beyond this there are always other important
desiderata of a less exact nature: The axioms should not be too numerous,
their system is to be as simple and transparent as possible, and each axiom
should have an immediate intuitive meaning by which its appropriateness
may be judged directly. 4 In a situation like ours this last requirement is
particularly vital, in spite of its vagueness: we want to make an intuitive
concept amenable to mathematical treatment and to see as clearly as
possible what hypotheses this requires.

The objective part of our problem is clear: the postulates must imply
the existence of a correspondence (3:2:a) with the properties (3:1 :a),
(3:l:b) as described in 3.5.1. The further heuristic, and even esthetic
desiderata, indicated above, do not determine a unique way of finding
this axiomatic treatment. In what follows we shall formulate a set of
axioms which seems to be essentially satisfactory.

1 I.e. a correspondence (3:2:a) which fulfills (3:1 :a), (3:1 :b).

8 I.e. one of the form (3:6).

3 Remember the physical examples of the same situation given in 3.4.4. (Our present
discussion is somewhat more detailed.) We do not undertake to fix an absolute zero
and an absolute unit of utility.

4 The first and the last principle may represent — at least to a certain extent — opposite
influences: If we reduce the number of axioms by merging them as far as technically
possible, we may lose the possibility of distinguishing the various intuitive backgrounds.
Thus we could have expressed the group (3:B) in 3.6.1. by a smaller number of axioms,
but this would have obscured the subsequent analysis of 3.6.2.

To strike a proper balance is a matter of practical — and to some extent even esthetic
— judgment.

26

FORMULATION OF THE ECONOMIC PROBLEM

3.6. The Axioms and Their Interpretation
3.6.1. Our axioms are these:

We consider a system U of entities 1 u, v y w, • • * . In V a relation is
given, u > v y and for any number a, (0 < a < 1), an operation

au + (1 — oi)v = w .

These concepts satisfy the following axioms:

(3: A) u > v is a complete ordering oj U . 2

This means: Write u < v when v > u. Then:

(3:A:a) For any two u , v one and only one of the three following

relations holds:

u = Vy u > v, u < v,

(3:A:b) u > v, v > w imply u > w. z

(3:B) Ordering and combining. 4

(3:B:a) u < v implies that u < au + (1 — a)v.

(3:B:b) u > v implies that u > au + (1 — a)v.

(3:B:c) u < w < v implies the existence of an a with

au + (1 — ot)v < w.

(3:B:d) u > w > v implies the existence of an a with

au + (1 — a)v > w.

(3:C) Algebra of combining.

(3:C:a) au + (1 — a)v = (1 — a)v + au.

(3:C:b) a(pu + (I — fi)v) + (1 — a)v = yu + (1 — y)v
where y = afi.

One can show that these axioms imply the existence of a correspondence
(3:2:a) with the properties (3:1 :a), (3:1 :b) as described in 3.5.1. Hence
the conclusions of 3.5.1. hold good: The system U — i.e. in our present
interpretation, the system of (abstract) utilities — is one of numbers up to
a linear transformation.

The construction of (3:2:a) (with (3:1 :a), (3:1 :b) by means of the
axioms (3:A)-(3:C)) is a purely mathematical task which is somewhat
lengthy, although it runs along conventional lines and presents no par-

1 This is, of course, meant to be the system of (abstract) utilities, to be characterized
by our axioms. Concerning the general nature of the axiomatic method, cf. the remarks
and references in the last part of 10.1.1.

*For a more systematic mathematical discussion of this notion, cf. 65.3.1. The
equivalent concept of the completeness of the system of preferences was previously con-
sidered at the beginning of 3.3.2. and of 3.4.6.

* These conditions (3:A:a), (3:A:b) correspond to (65:A:a), (65:A:b) in 65.3.1.

4 Remember that the a, 0, y occurring here are always >0, <1.

THE NOTION OF UTILITY

27

ticular difficulties. (Cf. Appendix.)

It seems equally unnecessary to carry out the usual logistic discussion
of these axioms 1 on this occasion.

We shall however say a few more words about the intuitive meaning —
i.e. the justification — of each one of our axioms (3:A)-(3:C).

3 . 6 . 2 . The analysis of our postulates follows:

(3:A:a*) This is the statement of the completeness of the system of
individual preferences. It is customary to assume this when
discussing utilities or preferences, e.g. in the “ indifference curve
analysis method.” These questions were already considered in
3.3.4. and 3.4.6.

(3:A:b*) This is the “ transitivity ” of preference, a plausible and
generally accepted property.

(3:B:a*) We state here: If v is preferable to u, then even a chance
1 — a of v — alternatively to u — is preferable. This is legitimate
since any kind of complementarity (or the opposite) has been
excluded, cf. the beginning of 3.3.2.

(3:B:b*) This is the dual of (3:B:a*), with “less preferable” in place of
“preferable.”

(3:B:c*) We state here: If w is preferable to u , and an even more

preferable v is also given, then the combination of u with a
chance 1 — a of v will not affect w’s preferability to it if this
chance is small enough. I.e.: However desirable v may be in
itself, one can make its influence as weak as desired by giving
it a sufficiently small chance. This is a plausible “continuity”
assumption.

(3:B:d*) This is the dual of (3:B:c*), with “less preferable” in place, of
“preferable.”

(3:C:a*) This is the statement that it is irrelevant in which order the
constituents u, v of a combination are named. It is legitimate,
particularly since the constituents are alternative events, cf.
(3:B:a*) above.

(3:C:b*) This is the statement that it is irrelevant whether a com-
bination of two constituents is obtained in two successive
steps, — first the probabilities a, 1 — a, then the probabilities 0,
1 — 0; or in one operation, — the probabilities y, 1 — y where
7 = a/3. 2 The same things can be said for this as for (3:C:a*)
above. It may be, however, that this postulate has a deeper
significance, to which one allusion is made in 3.7.1. below.

1 A similar situation is dealt with more exhaustively in 10.; those axioms describe a
subject which is more vital for our main objective. The logistic discussion is indicated
there in 10.2. Some of the general remarks of 10.3. apply to the present case also.

2 This is of course the correct arithmetic of accounting for two successive admixtures
of v with u.

28

FORMULATION OF THE ECONOMIC PROBLEM

3.7. General Remarks Concerning the Axioms

3 . 7 . 1 . At this point it may be well to stop and to reconsider the situa-
tion. Have we not shown too much? We can derive from the postulates
(3:A)-(3:C) the numerical character of utility in the sense of (3:2:a) and
(3:1 :a), (3:1 :b) in 3.5.1.; and (3:1 :b) states that the numerical values of
utility combine (with probabilities) like mathematical expectations! And
yet the concept of mathematical expectation has been often questioned,
and its legitimateness is certainly dependent upon some hypothesis con-
cerning the nature of an “expectation.” 1 Have we not then begged the
question? Do not our postulates introduce, in some oblique way, the
hypotheses which bring in the mathematical expectation?

More specifically: May there not exist in an individual a (positive or
negative) utility of the mere act of “taking a chance,” of gambling, which
the use of the mathematical expectation obliterates?

How did our axioms (3:A)-(3:C) get around this possibility?

As far as we can see, our postulates (3:A)-(3:C) do not attempt to avoid
it. Even that one which gets closest to excluding a “utility of gambling”
(3:C:b) (cf. its discussion in 3.6.2.), seems to be plausible and legitimate, —
unless a much more refined system of psychology is used than the one now
available for the purposes of economics. The fact that a numerical utility —
with a formula amounting to the use of mathematical expectations — can
be built upon (3:A)-(3:C), seems to indicate this: We have practically
defined numerical utility as being that thing for which the calculus of
mathematical expectations is legitimate. 2 Since (3:A)-(3:C) secure that
the necessary construction can be carried out, concepts like a “specific
utility of gambling” cannot be formulated free of contradiction on this
level. 3

3 . 7 . 2 . As we have stated, the last time in 3.6.1., our axioms are based
on the relation u > v and on the operation au + (1 — a)v for utilities.
It seems noteworthy that the latter may be regarded as more immediately
given than the former: One can hardly doubt that anybody who could
imagine two alternative situations with the respective utilities u , v could
not also conceive the prospect of having both with the given respective
probabilities a, 1 — a. On the other hand one may question the postulate
of axiom (3:A:a) for u > v, i.e. the completeness of this ordering.

Let us consider this point for a moment. We have conceded that one
may doubt whether a person'Jcan always decide which of two alternatives —

1 Cf. Karl Menger: Das Unsicherheitsmoment in der Wertlehre, Zeitschrift fiir
Nationalfikonomie, vol. 5, (1934) pp. 459ff. and Gerhard Tintner: A contribution to the
non-static Theory of Choice, Quarterly Journal of Economics, vol. LVI, (1942) pp. 274ff.

•Thus Daniel Bernoulli’s well known suggestion to “ solve” the “St. Petersburg
Paradox” by the use of the so-called “moral expectation” (instead of the mathematical
expectation) means defining the utility numerically as the logarithm of one’s monetary
possessions.

1 This may seem to be a paradoxical assertion. But anybody who has seriously tried
to axiomatize that elusive concept, will probably concur with it.

THE NOTION OF UTILITY

29

with the utilities u } v — he prefers. 1 But, whatever the merits of this
doubt are, this possibility — i.e. the completeness of the system of (indi-
vidual) preferences — must be assumed even for the purposes of the “indiffer-
ence curve method” (cf. our remarks on (3:A:a) in 3.6.2.). But if this
property of u > v 2 is assumed, then our use of the much less questionable
au + (1 — a)v 8 yields the numerical utilities too! 4

If the general comparability assumption is not made, 6 a mathematical
theory — based on au + (1 — a)v together with what remains of u > v —
is still possible. 8 It leads to what may be described as a many-dimensional
vector concept of utility. This is a more complicated and less satisfactory
set-up, but we do not propose to treat it systematically at this time.

3 . 7 . 3 . This brief exposition does not claim to exhaust the subject, but
we hope to have conveyed the essential points. To avoid misunderstand-
ings, the following further remarks may be useful.

(1) We re-emphasize that we are considering only utilities experienced
by one person. These considerations do not imply anything concerning the
comparisons of the utilities belonging to different individuals.

(2) It cannot be denied that the analysis of the methods which make use
of mathematical expectation (cf. footnote 1 on p. 28 for the literature) is
far from concluded at present. Our remarks in 3.7.1. lie in this direction,
but much more should be said in this respect. There are many interesting
questions involved, which however lie beyond the scope of this work.
For our purposes it suffices to observe that the validity of the simple and
plausible axioms (3:A)-(3:C) in 3.6.1. for the relation u > v and the oper-
ation au + (1 — a) v makes the utilities numbers up to a linear transforma-
tion in the sense discussed in these sections.

3.8. The Role of the Concept of Marginal Utility .

3 . 8 . 1 . The preceding analysis made it clear that we feel free to make
use of a numerical conception of utility. On the other hand, subsequent

1 Or that he can assert that they are precisely equally desirable.

2 I.e. the completeness postulate (3:A:a).

* I.e. the postulates (3:B), (3:C) together with the obvious postulate (3:A:b).

4 At this point the reader may recall the familiar argument according to which the
unnumerical (“indifference curve”) treatment of utilities is preferable to any numerical
one, because it is simpler and based on fewer hypotheses. This objection might be
legitimate if the numerical treatment were based on Pareto’s equality relation for utility
differences (cf. the end of 3.4.6.). This relation is, indeed, a stronger and more compli-
cated hypothesis, added to the original ones concerning the general comparability of
utilities (completeness of preferences).

However, we used the operation au - b (1 — a)t> instead, and we hope that the reader
will agree with us that it represents an even safer assumption than that of the complete-
ness of preferences.

We think therefore that our procedure, as distinguished from Pareto’s, is not open
to the objections based on the necessity of artificial assumptions and a loss of simplicity.

6 This amounts to weakening (3:A:a) to an (3:A:a') by replacing in it “one and only
one” by “at most one.” The conditions (3:A:a')> (3:A:b) then correspond to (65:B:a),
(65:B:b).

•In this case some modifications in the groups of postulates (3:B), (3:C) are also
necessary.

30

FORMULATION OF THE ECONOMIC PROBLEM

discussions will show that we cannot avoid the assumption that all subjects
of the economy under consideration are completely informed about the
physical characteristics of the situation in which they operate and are able
to perform all statistical, mathematical, etc., operations which this knowl-
edge makes possible. The nature and importance of this assumption has
been given extensive attention in the literature and the subject is probably
very far from being exhausted. We propose not to enter upon it. The
question is too vast and too difficult and we believe that it is best to “divide
difficulties.” I.e. we wish to avoid this complication which, while interest-
ing in its own right, should be considered separately from our present
problem.

Actually we think that our investigations — although they assume
“complete information” without any further discussion — do make a con-
tribution to the study of this subject. It will be seen that many economic
and social phenomena which are usually ascribed to the individual's state of
“incomplete information” make their appearance in our theory and can be
satisfactorily interpreted with its help. Since our theory assumes “com-
plete information,” we conclude from this that those phenomena have
nothing to do with the individual's “incomplete information.” Some
particularly striking examples of this will be found in the concepts of
“discrimination” in 33.1., of “incomplete exploitation” in 38.3., and of the
“transfer” or “tribute” in 46.11., 46.12.

On the basis of the above we would even venture to question the impor-
tance usually ascribed to incomplete information in its conventional sense 1
in economic and social theory. It will appear that some phenomena which
would prima facie have to be attributed to this factor, have nothing to do
with it. 2

3 . 8 . 2 . Let us now consider an isolated individual with definite physical
characteristics and with definite quantities of goods at his disposal. In
view of what was said above, he is in a position to determine the maximum
utility which can be obtained in this situation. Since the maximum is a
well-defined quantity, the same is true for the increase which occurs when a
unit of any definite good is added to the stock of all goods in the possession
of the individual. This is, of course, the classical notion of the marginal
utility of a unit of the commodity in question. 3

These quantities are clearly of decisive importance in the “Robinson
Crusoe” economy. The above marginal utility obviously corresponds to

1 We shall see that the rules of the games considered may explicitly prescribe that
certain participants should not possess certain pieces of information. Cf. 6.3., 6.4.
(Games in which this does not happen are referred to in 14.8. and in (15:B) of 15.3.2., and
are called games with “ perfect information. ”) We shall recognize and utilize this kind of
“incomplete information ” (according to the above, rather to be called “imperfect
information ,, ). But we reject all other types, vaguely defined by the use of concepts
like complication, intelligence, etc.

2 Our theory attributes these phenomena to the possibility of multiple “stable
standards of behavior” cf 4.6. and the end of 4.7.

* More precisely: the so-called “indirectly dependent expected utility.”

SOLUTIONS AND STANDARDS OF BEHAVIOR 31

the maximum effort which he will be willing to make — if he behaves accord-
ing to the customary criteria of rationality — in order to obtain a further
unit of that commodity.

It is not clear at all, however, what significance it has in determining
the behavior of a participant in a social exchange economy. We saw that
the principles of rational behavior in this case still await formulation, and
that they are certainly not expressed by a maximum requirement of the
Crusoe type. Thus it must be uncertain whether marginal utility has any
meaning at all in this case. 1

Positive statements on this subject will be possible only after we have
succeeded in developing a theory of rational behavior in a social exchange
economy, — that is, as was stated before, with the help of the theory of
“games of strategy.” It will be seen that marginal utility does, indeed,
play an important role in this case too, but in a more subtle way than is
usually assumed.

4. Structure of the Theory : Solutions and Standards of Behavior

4.1. The Simplest Concept of a Solution for One Participant

4.1.1. We have now reached the point where it becomes possible to
give a positive description of our proposed procedure. This means pri-
marily an outline and an account of the main technical concepts and
devices.

As we stated before, we wish to find the mathematically complete
principles which define “rational behavior” for the participants in a social
economy, and to derive from them the general characteristics of that
behavior. And while the principles ought to be perfectly general — i.e. ,
valid in all situations — we may be satisfied if we can find solutions, for the
moment, only in some characteristic special cases.

First of all we must obtain a clear notion of what can be accepted as a
solution of this problem; i.e., what the amount of information is which a
solution must convey, and what we should expect regarding its formal
structure. A precise analysis becomes possible only after these matters
have been clarified.

4.1.2. The immediate concept of a solution is plausibly a set of rules for
each participant which tell him how to behave in every situation which may
conceivably arise. One may object at this point that this view is unneces-
sarily inclusive. Since we want to theorize about “ rational behavior,” there
seems to be no need to give the individual advice as to his behavior in
situations other than those which arise in a rational community. This
would justify assuming rational behavior on the part of the others as well, —
in whatever way we are going to characterize that. Such a procedure
would probably lead to a unique sequence of situations to which alone our
theory need refer.

1 All this is understood within the domain of our several simplifying assumptions. If
they are relaxed, then various further difficulties ensue.

32

FORMULATION OF THE ECONOMIC PROBLEM

This objection seems to be invalid for two reasons:

First, the “ rules of the game,” — i.e. the physical laws which give the
factual background of the economic activities under consideration may be
explicitly statistical. The actions of the participants of the economy may
determine the outcome only in conjunction with events which depend on
chance (with known probabilities), cf. footnote 2 on p. 10 and 6.2.1. If
this is taken into consideration, then the rules of behavior even in a perfectly
rational community must provide for a great variety of situations — some of
which will be very far from optimum. 1

Second, and this is even more fundamental, the rules of rational behavior
must provide definitely for the possibility of irrational conduct on the part
of others. In other words: Imagine that we have discovered a set of rules
for all participants — to be termed as “ optimal” or “ rational” — each of
which is indeed optimal provided that the other participants conform.
Then the question remains as to what will happen if some of the participants
do not conform. If that should turn out to be advantageous for them — and,
quite particularly, disadvantageous to the conformists — then the above
“solution” would seem very questionable. We are in no position to give a
positive discussion of these things as yet — but we want to make it clear
that under such conditions the “solution,” or at least its motivation, must
be considered as imperfect and incomplete. In whatever way we formulate
the guiding principles and the objective justification of “rational behavior,”
provisos will have to be made for every possible conduct of “the others.”
Only in this way can a satisfactory and exhaustive theory be developed.
But if the superiority of “rational behavior” over any other kind is to be
established, then its description must include rules of conduct for all
conceivable situations — including those where “the others” behaved
irrationally, in the sense of the standards which the theory will set for them.

4 . 1 , 3 . At this stage the reader will observe a great similarity with the
everyday concept of games. We think that this similarity is very essential;
indeed, that it is more than that. For economic and social problems the
games fulfill — or should fulfill — the same function which various geometrico-
mathematical models have successfully performed in the physical sciences.
Such models are theoretical constructs with a precise, exhaustive and not
too complicated definition; and they must be similar to reality in those
respects which are essential in the investigation at hand. To reca-
pitulate in detail: The definition must be precise and exhaustive in
order to make a mathematical treatment possible. The construct must
not be unduly complicated, so that the mathematical treatment can be
brought beyond the mere formalism to the point where it yields complete
numerical results. Similarity to reality is needed to make the operation
significant. And this similarity must usually be restricted to a few traits

1 That a unique optimal behavior is at all conceivable in spite of the multiplicity of
the possibilities determined by chance, is of course due to the use of the notion of “ mathe-
matical expectation/’ Cf. loc. cit. above.

SOLUTIONS AND STANDARDS OF BEHAVIOR

33

deemed “essential” pro tempore — since otherwise the above requirements
would conflict with each other. 1

It is clear that if a model of economic activities is constructed according
to these principles, the description of a game results. This is particularly
striking in the formal description of markets which are after all the core
of the economic system — but this statement is true in all cases and without
qualifications.

4 . 1 . 4 . We described in 4.1.2. what we expect a solution — i.e. a character-
ization of “ rational behavior ” — to consist of. This amounted to a complete
set of rules of behavior in all conceivable situations. This holds equiv-
alently for a social economy and for games. The entire result in the
above sense is thus a combinatorial enumeration of enormous complexity.
But we have accepted a simplified concept of utility according to which all
the individual strives for is fully described by one numerical datum (cf.
2.1.1. and 3.3.). Thus the complicated combinatorial catalogue — which
we expect from a solution — permits a very brief and significant summariza-
tion: the statement of how much 2 - 3 the participant under consideration can
get if he behaves “rationally.” This “can get” is, of course, presumed to
be a minimum; he may get more if the others make mistakes (behave
irrationally).

It ought to be understood that all this discussion is advanced, as it
should be, preliminary to the building of a satisfactory theory along the
lines indicated. We formulate desiderata which will serve as a gauge of
success in our subsequent considerations; but it is in accordance with the
usual heuristic procedure to reason about these desiderata — even before
we are able to satisfy them. Indeed, this preliminary reasoning is an
essential part of the process of finding a satisfactory theory. 4

4.2. Extension to All Participants

4 . 2 . 1 . We have considered so far only what the solution ought to be for
one participant. Let us now visualize all participants simultaneously.
I.e., let us consider a social economy, or equivalently a game of a fixed
number of (say n) participants. The complete information which a solution
should convey is, as we discussed it, of a combinatorial nature. It was
indicated furthermore how a single quantitative statement contains the
decisive part of this information, by stating how much each participant

1 E.g., Newton’s description of the solar system by a small number of “masspoints.”
These points attract each other and move like the stars ; this is the similarity in the essen-
tials, while the enormous wealth of the other physical features of the planets has been left
out of account.

2 Utility; for an entrepreneur, — profit; for a player, — gain or loss.

* We mean, of course, the “mathematical expectation/’ if there is an explicit element
of chance. Cf. the first remark in 4.1.2. and also the discussion of 3.7.1.

4 Those who are familiar with the development of physics will know how important
such heuristic considerations can be. Neither general relativity nor quantum mechanics
could have been found without a “nrA_thpnrptin«l ,/ dinniiosinn of t.hft desiderata concern-
ing the theory-to-be.

34 FORMULATION OF THE ECONOMIC PROBLEM

obtains by behaving rationally. Consider these amounts which the several
participants “obtain.” If the solution did nothing more in the quantitative
sense than specify these amounts, 1 then it would coincide with the well
known concept of imputation: it would just state how the total proceeds
are to be distributed among the participants. 2

We emphasize that the problem of imputation must be solved both
when the total proceeds are in fact identically zero and when they are vari-
able. This problem, in its general form, has neither been properly formu-
lated nor solved in economic literature.

4.2.2. We can see no reason why one should not be satisfied with a
solution of this nature, providing it can be found: i.e. a single imputation
which meets reasonable requirements for optimum (rational) behavior.
(Of course we have not yet formulated these requirements. For an exhaus-
tive discussion, cf. loc. cit. below.) The structure of the society under con-
sideration would then be extremely simple: There would exist an absolute
state of equilibrium in which the quantitative share of every participant
would be precisely determined.

It will be seen however that such a solution, possessing all necessary
properties, does not exist in general. The notion of a solution will have
to be broadened considerably, and it will be seen that this is closely con-
nected with certain inherent features of social organization that are well
known from a “common sense” point of view but thus far have not been
viewed in proper perspective. (Cf. 4.6. and 4.8.1.)

4.2.3. Our mathematical analysis of the problem will show that there
exists, indeed, a not inconsiderable family of games where a solution can be
defined and found in the above sense: i.e. as one single imputation. In
such cases every participant obtains at least the amount thus imputed to
him by just behaving appropriately, rationally. Indeed, he gets exactly
this amount if the other participants too behave rationally; if they do not,
he may get even more.

These are the games of two participants where the sum of all payments
is zero. While these games are not exactly typical for major economic
processes, they contain some universally important traits of all games and
the results derived from them are the basis of the general theory of games.
We shall discuss them at length in Chapter III.

4.8. The Solution as a Set of Imputations

4.3.1. If either of the two above restrictions is dropped, the situation is
altered materially.

1 And of course, in the combinatorial sense, as outlined above, the procedure how to
obtain them.

* In games — as usually understood — the total proceeds are always zero; i.e. one
participant can gain only what the others lose. Thus there is a pure problem of distri-
bution — i.e. imputation — and absolutely none of increasing the total utility, the “social
product.” In all economic questions the latter problem arises as well, but the question
of imputation remains. Subsequently we shall broaden the concept of a game by drop-
ping the requirement of the total proceeds being zero (cf. Ch. XI).

SOLUTIONS AND STANDARDS OF BEHAVIOR

35

The simplest game where the second requirement is overstepped is a
two-person game where the sum of all payments is variable. This cor-
responds to a social economy with two participants and allows both for
their interdependence and for variability of total utility with their behavior. 1
As a matter of fact this is exactly the case of a bilateral monopoly (cf.
6L2.-61.6.). The well known “zone of uncertainty” which is found in
current efforts to solve the problem of imputation indicates that a broader
concept of solution must be sought. This case will be discussed loc. cit.
above. For the moment we want to use it only as an indicator of the diffi-
culty and pass to the other case which is more suitable as a basis for a first
positive step.

4.3.2. The simplest game where the first requirement is disregarded is a
three-person game where the sum of all payments is zero. In contrast to
the above two-person game, this does not correspond to any fundamental
economic problem but it represents nevertheless a basic possibility in human
relations. The essential feature is that any two players who combine and
cooperate against a third can thereby secure an advantage. The problem
is how this advantage should be distributed among the two partners in this
combination. Any such scheme of imputation will have to take into
account that any two partners can combine; i.e. while any one combination
is in the process of formation, each partner must consider the fact that his
prospective ally could break away and join the third participant.

Of course the rules of the game will prescribe how the proceeds of a
coalition should be divided between the partners. But the detailed dis-
cussion to be given in 22.1. shows that this will not be, in general, the
final verdict. Imagine a game (of three or more persons) in which two
participants can form a very advantageous coalition but where the rules
of the game provide that the greatest part of the gain goes to the first
participant. Assume furthermore that the second participant of this
coalition can also enter a coalition with the third one, which is less effective
in toto but promises him a greater individual gain than the former. In
this situation it is obviously reasonable for the first participant to transfer
a part of the gains which he could get from the first coalition to the second
participant in order to save this coalition. In other words: One must
expect that under certain conditions one participant of a coalition will be
willing to pay a compensation to his partner. Thus the apportionment
within a coalition depends not only upon the rules of the game but
also upon the above principles, under the influence of the alternative
coalitions. 2

Common sense suggests that one cannot expect any theoretical state-
ment as to which alliance will be formed 3 but only information concerning

1 It will be remembered that we make use of a transferable utility, cf. 2.1.1.

* This does not mean that the rules of the game are violated, since such compensatory
payments, if made at all, are made freely in pursuance of a rational consideration.

1 Obviously three combinations of two partners each are possible. In the example
to be given in 21., any preference within the solution for a particular alliance will be a

36

FORMULATION OF THE ECONOMIC PROBLEM

how the partners in a possible combination must divide the spoils in order
to avoid the contingency that any one of them deserts to form a combination
with the third player. All this will be discussed in detail and quantitatively
in Ch. V.

It suffices to state here only the result which the above qualitative
considerations make plausible and which will be established more rigorously
loc. cit. A reasonable concept of a solution consists in this case of a system
of three imputations. These correspond to the above-mentioned three
combinations or alliances and express the division of spoils between respec-
tive allies.

4 . 3 . 3 . The last result will turn out to be the prototype of the general
situation. We shall see that a consistent theory will result from looking
for solutions which are not single imputations, but rather systems of
imputations.

It is clear that in the above three-person game no single imputation
from the solution is in itself anything like a solution. Any particular
alliance describes only one particular consideration which enters the minds
of the participants when they plan their behavior. Even if a particular
alliance is ultimately formed, the division of the proceeds between the allies
will be decisively influenced by the other alliances which each one might
alternatively have entered. Thus only the three alliances and their
imputations together form a rational whole which determines all of its
details and possesses a stability of its own. It is, indeed, this whole which
is the really significant entity, more so than its constituent imputations.
Even if one of these is actually applied, i.e. if one particular alliance is
actually formed, the others are present in a “virtual” existence: Although
they have not materialized, they have contributed essentially to shaping and
determining the actual reality.

In conceiving of the general problem, a social economy or equivalently
a game of n participants, we shall — with an optimism which can be justified
only by subsequent success — expect the same thing: A solution should be a
system of imputations 1 possessing in its entirety some kind of balance and
stability the nature of which we shall try to determine. We emphasize
that this stability — whatever it may turn out to be — will be a property
of the system as a whole and not of the single imputations of which it is
composed. These brief considerations regarding the three-person game
have illustrated this point.

4 . 3 . 4 . The exact criteria which characterize a system of imputations as a
solution of our problem are, of course, of a mathematical nature. For a
precise and exhaustive discussion we must therefore refer the reader to the
subsequent mathematical development of the theory. The exact definition

limine excluded by symmetry. I.e. the game will be symmetric with respect to all three
participants. Cf. however 33.1.1.

1 They may again include compensations between partners in a coalition, as described
in 4.3.2.

SOLUTIONS AND STANDARDS OF BEHAVIOR 37

itself is stated in 30.1.1. We shall nevertheless undertake to give a prelimi-
nary, qualitative outline. We hope this will contribute to the understanding
of the ideas on which the quantitative discussion is based. Besides, the
place of our considerations in the general framework of social theory will
become clearer.

4.4. The Intransitive Notion of 41 Superiority” or 41 ‘Domination”

4 . 4 . 1 . Let us return to a more primitive concept of the solution which we
know already must be abandoned. We mean the idea of a solution as a
single imputation. If this sort of solution existed it would have to be an
imputation which in some plausible sense was superior to all other imputa-
tions. This notion of superiority as between imputations ought to be
formulated in a way which takes account of the physical and social struc-
ture of the milieu. That is, one should define that an imputation x is
superior to an imputation y whenever this happens: Assume that society,
i.e. the totality of all participants, has to consider the question whether or
not to “accept” a static settlement of all questions of distribution by the
imputation y. Assume furthermore that at this moment the alternative
settlement by the imputation x is also considered. Then this alternative x
will suffice to exclude acceptance of y . By this we mean that a sufficient
number of participants prefer in their own interest x to y , and are convinced
or can be convinced of the possibility of obtaining the advantages of x.
In this comparison of x to y the participants should not be influenced by
the consideration of any third alternatives (imputations). I.e. we conceive
the relationship of superiority as an elementary one, correlating the two
imputations x and y only. The further comparison of three or more —
ultimately of all — imputations is the subject of the theory which must
now follow, as a superstructure erected upon the elementary concept of
superiority.

Whether the possibility of obtaining certain advantages by relinquishing
y for x y as discussed in the above definition, can be made convincing to the
interested parties will depend upon the physical facts of the situation — in
the terminology of games, on the rules of the game.

We prefer to use, instead of “superior” with its manifold associations, a
word more in the nature of a terminus technicus. When the above described
relationship between two imputations x and y exists, 1 then we shall say
that x dominates y.

If one restates a little more carefully what should be expected from a
solution consisting of a single imputation, this formulation obtains: Such
an imputation should dominate all others and be dominated by
none.

4 . 4 . 2 . The notion of domination as formulated — or rather indicated —
above is clearly in the nature of an ordering, similar to the question of

1 That is, when it holds in the mathematically precise form, which will be given in
30.1.1.

38

FORMULATION OF THE ECONOMIC PROBLEM

preference, or of size in any quantitative theory. The notion of a single
imputation solution 1 corresponds to that of the first element with respect
to that ordering. 2

The search for such a first element would be a plausible one if the order-
ing in question, i.e. our notion of domination, possessed the important
property of transitivity; that is, if it were true that whenever x dominates
y and y dominates z , then also x dominates z. In this case one might proceed
as follows: Starting with an arbitrary x, look for a y which dominates x; if
such a y exists, choose one and look for a z which dominates y ; if such a z
exists, choose one and look for a u which dominates z, etc. In most practical
problems there is a fair chance that this process either terminates after a
finite number of steps with a w which is undominated by anything else, or
that the sequence x, y, z, u, • • • , goes on ad infinitum , but that these
x, y, z, u y • • • tend to a limiting position w undominated by anything else.
And, due to the transitivity referred to above, the final w will in either case
dominate all previously obtained x, y y z, u, • • • .

We shall not go into more elaborate details which could and should
be given in an exhaustive discussion. It will probably be clear to the reader
that the progress through the sequence x, y, z, u, • • • corresponds to
successive “ improvements ’ ’ culminating in the “optimum,” i.e. the “first”
element w which dominates all others and is not dominated.

All this becomes very different when transitivity does not prevail.
In that case any attempt to reach an “optimum” by successive improve-
ments may be futile. It can happen that x is dominated by y, y by z, and
z in turn by x. 8

4,4.3. Now the notion of domination on which we rely is, indeed, not
transitive. In our tentative description of this concept we indicated that x
dominates y when there exists a group of participants each one of whom
prefers his individual situation in x to that in y , and who are convinced
that they are able as a group — i.e. as an alliance — to enforce their prefer-
ences. We shall discuss these matters in detail in 30.2. This group of
participants shall be called the “effective set” for the domination of x over y.
Now when x dominates y and y dominates z, the effective sets for these two
dominations may be entirely disjunct and therefore no conclusions can be
drawn concerning the relationship between z and x. It can even happen
that z dominates x with the help of a third effective set, possibly disjunct'
from both previous ones.

1 We continue to use it as an illustration although we have shown already that it is a
forlorn hope. The reason for this is that, by showing what is involved if certain complica-
tions did not arise, we can put these complications into better perspective. Our real
interest at this stage lies of course in these complications, which are quite fundamental.

1 The mathematical theory of ordering is very simple and leads probably to a deeper
understanding of these conditions than any purely verbal discussion. The necessary
mathematical considerations will be found in 65.3.

3 In the case of transitivity this is impossible because — if a proof be wanted — x never
dominates itself. Indeed, if e.g. y dominates z, z dominates y } and x dominates z, then
we can infer by transitivity that x dominates x.

SOLUTIONS AND STANDARDS OF BEHAVIOR

39

This lack of transitivity, especially in the above formalistic presentation,
may appear to be an annoying complication and it may even seem desirable
to make an effort to rid the theory of it. Yet the reader who takes another
look at the last paragraph will notice that it really contains only a circum-
locution of a most typical phenomenon in all social organizations. The
domination relationships between various imputations x, y, z, • • • — i.e.
between various states of society — correspond to the various ways in which
these can unstabilize — i.e. upset — each other. That various groups of
participants acting as effective sets in various relations of this kind may
bring about “cyclical” dominations — e.g., y over x , z over y , and x over z —
is indeed one of the most characteristic difficulties which a theory of these
phenomena must face.

4.5. The Precise Definition of a Solution

4 . 6 . 1 . Thus our task is to replace the notion of the optimum — i.e. of the
first element — by something which can take over its functions in a static
equilibrium. This becomes necessary because the original concept has
become untenable. We first observed its breakdown in the specific instance
of a certain three-person game in 4.3. 2. -4. 3. 3. But now we have acquired
a deeper insight into the cause of its failure: it is the nature of our concept of
domination, and specifically its intransitivity.

This type of relationship is not at all peculiar to our problem. Other
instances of it are well known in many fields and it is to be regretted that
they have never received a generic mathematical treatment. We mean all
those concepts which are in the general nature of a comparison of preference
or “superiority,” or of order, but lack transitivity: e.g., the strength of
chess players in a tournament, the “paper form” in sports and races, etc. 1

4 . 6 . 2 . The discussion of the three-person game in 4.3.2.-4.3.3. indicated
that the solution will be, in general, a set of imputations instead of a single
imputation. That is, the concept of the “first element” will have to be
replaced by that of a set of elements (imputations) with suitable properties.
In the exhaustive discussion of this game in 32. (cf. also the interpreta-
tion in 33.1.1. which calls attention to some deviations) the system of three
imputations, which was introduced as the solution of the three-person game in

4.3.2.-4.3.3., will be derived in an exact way with the help of the postulates
of 30.1.1. These postulates will be very similar to those which character-
ize a first element. They are, of course, requirements for a set of elements
(imputations), but if that set should turn out to consist of a single element
only, then our postulates go over into the characterization of the first
element (in the total system of all imputations).

We do not give a detailed motivation for those postulates as yet, but we
shall formulate them now hoping that the reader will find them to be some-

1 Some of these problems have been treated mathematically by the introduction of
chance and probability. Without denying that this approach has a certain justification,
we doubt whether it is conducive to a complete understanding even in those cases. It
would be altogether inadequate for our considerations of social organization.

40

FORMULATION OF THE ECONOMIC PROBLEM

what plausible. Some reasons of a qualitative nature, or rather one possible
interpretation, will be given in the paragraphs immediately following.

4 . 6 . 3 . The postulates are as follows: A set S of elements (imputations)
is a solution when it possesses these two properties:

(4:A:a) No y contained in S is dominated by an x contained in S.

(4:A:b) Every y not contained in S is dominated by some x con-

tained in S.

(4:A:a) and (4:A:b) can be stated as a single condition:

(4:A:c) The elements of S are precisely those elements which are

undominated by elements of S. 1

The' reader who is interested in this type of exercise may now verify
our previous assertion that for a set S which consists of a single element x
the above conditions express precisely that x is the first element.

4 . 6 . 4 . Part of the malaise which the preceding postulates may cause at
first sight is probably due to their circular character. This is particularly
obvious in the form (4:A:c), where the elements of S are characterized by a
relationship which is again dependent upon S. It is important not to
misunderstand the meaning of this circumstance.

Since our definitions (4:A:a) and (4:A:b), or (4:A:c), are circular — i.e.
implicit — for S, it is not at all clear that there really exists an S which
fulfills them, nor whether — if there exists one — the S is unique. Indeed
these questions, at this stage still unanswered, are the main subject of the
subsequent theory. What is clear, however, is that these definitions tell
unambiguously whether any particular S is or is not a solution. If one
insists on associating with the concept of a definition the attributes of
existence and uniqueness of the object defined, then one must say: We
have not given a definition of S , but a definition of a property of S — we
have not defined the solution but characterized all possible solutions.
Whether the totality of all solutions, thus circumscribed, contains no S ,
exactly one S , or several S’ s, is subject for further inquiry. 2

4.6. Interpretation of Our Definition in Terms of “Standards of Behavior”

4 . 6 . 1 . The single imputation is an often used and well understood con-
cept of economic theory, while the sets of imputations to which we have
been led are rather unfamiliar ones. It is therefore desirable to correlate
them with something which has a well established place in our thinking
concerning social phenomena.

1 Thus (4: Arc) is an exact equivalent of (4:A:a) and (4:A:b) together. It may impress
the mathematically untrained reader as somewhat involved, although it is really a
straightforward expression of rather simple ideas.

2 It should be unnecessary to say that the circularity, or rather implicitness, of
(4:A:a) and (4:A:b), or (4:A:c), does not at all mean that they are tautological. They
express, of course, a very serious restriction of S .

SOLUTIONS AND STANDARDS OF BEHAVIOR

41

Indeed, it appears that the sets of imputations S which we are consider-
ing correspond to the “ standard of behavior” connected with a social
organization. Let us examine this assertion more closely.

Let the physical basis of a social economy be given, — or, to take a
broader view of the matter, of a society. 1 According to all tradition and
experience human beings have a characteristic way of adjusting themselves
to such a background. This consists of not setting up one rigid system of
apportionment, i.e. of imputation, but rather a variety of alternatives,
which will probably all express some general principles but nevertheless
differ among themselves in many particular respects. 2 This system of
imputations describes the “ established order of society” or “ accepted
standard of behavior.”

Obviously no random grouping of imputations will do as such a “ stand-
ard of behavior”: it will have to satisfy certain conditions which character-
ize it as a possible order of things. This concept of possibility must clearly
provide for conditions of stability. The reader will observe, no doubt,
that our procedure in the previous paragraphs is very much in this spirit:
The sets S of imputations x y y, z, • • • correspond to what we now call
“standard of behavior,” and the conditions (4:A:a) and (4:A:b), or (4:A:c),
which characterize the solution S express, indeed, a stability in the above
sense.

4 . 6 . 2 . The disjunction into (4:A:a) and (4:A:b) is particularly appropri-
ate in this instance. Recall that domination of y by x means that the
imputation x, if taken into consideration, excludes acceptance of the
imputation y (this without forecasting what imputation will ultimately be
accepted, cf. 4.4.1. and 4.4.2.). Thus (4:A:a) expresses the fact that the
standard of behavior is free from inner contradictions: No imputation y
belonging to S — i.e. conforming with the “accepted standard of behavior”
— can be upset — i.e. dominated — by another imputation x of the same kind.
On the other hand (4:A:b) expresses that the “standard of behavior” can
be used to discredit any non-conforming procedure: Every imputation y
not belonging to S can be upset — i.e. dominated — by an imputation x
belonging to S.

Observe that we have not postulated in 4.5.3. that a y belonging to S
should never be dominated by any x. 3 Of course, if this should happen, then
x would have to be outside of S, due to (4:A:a). In the terminology of
social organizations: An imputation y which conforms with the “accepted

1 In the case of a game this means simply — as we have mentioned before — that the
rules of the game are given. But for the present simile the comparison with a social
economy is more useful. We suggest therefore that the reader forget temporarily the
analogy with games and think entirely in terms of social organization.

2 There may be extreme, or to use a mathematical term, “degenerate” special cases
where the setup is of such exceptional simplicity that a rigid single apportionment can
be put into operation. But it seems legitimate to disregard them as non-typical.

3 It can be shown, cf. (31 :M) in 31.2.3., that such a postulate cannot be fulfilled
in general ; i.e. that in all really interesting cases it is impossible to find an S which satisfies
it together with our other requirements.

42

FORMULATION OF THE ECONOMIC PROBLEM

standard of behavior” may be upset by another imputation x, but in this
case it is certain that x does not conform. 1 It follows from our other require-
ments that then x is upset in turn by a third imputation z which again
conforms. Since y and z both conform, z cannot upset y — a further illustra-
tion of the intransitivity of “domination.”

Thus our solutions S correspond to such “standards of behavior ’ as
have an inner stability: once they are generally accepted they overrule
everything else and no part of them can be overruled within the limits of
the accepted standards. This is clearly how things are in actual social
organizations, and it emphasizes the perfect appropriateness of the circular
character of our conditions in 4.5.3.

4 . 6 . 3 . We have previously mentioned, but purposely neglected to dis-
cuss, an important objection: That neither the existence nor the uniqueness
of a solution S in the sense of the conditions (4:A:a) and (4:A:b), or (4:A:c),
in 4.5.3. is evident or established.

There can be, of course, no concessions as regards existence. If it
should turn out that our requirements concerning a solution S are, in any
special case, unfulfillable, — this would certainly necessitate a fundamental
change in the theory. Thus a general proof of the existence of solutions aS
for all particular cases 2 is most desirable. It will appear from our subse-
quent investigations that this proof has not yet been carried out in full
generality but that in all cases considered so far solutions were found.

As regards uniqueness the situation is altogether different. The often
mentioned “circular” character of our requirements makes it rather
probable that the solutions are not in general unique. Indeed we shall in
most cases observe a multiplicity of solutions. 3 Considering what we have
said about interpreting solutions as stable “standards of behavior” this has
a simple and not unreasonable meaning, namely that given the same
physical background different “established orders of society” or “accepted
standards of behavior” can be built, all possessing those characteristics of
inner stability which we have discussed. Since this concept of stability
is admittedly of an “inner” nature — i.e. operative only under the hypothesis
of general acceptance of the standard in question — these different standards
may perfectly well be in contradiction with each other.

4 . 6 . 4 . Our approach should be compared with the widely held view
that a social theory is possible only on the basis of some preconceived
principles of social purpose. These principles would include quantitative
statements concerning both the aims to be achieved in toto and the appor-
tionments between individuals. Once they are accepted, a simple maximum
problem results.

1 We use the word “conform” (to the “standard of behavior”) temporarily as a
synonym for being contained in S , and the word “upset” as a synonym for dominate.

*In the terminology of games: for all numbers of participants and for all possible
rules of the game.

1 An interesting exception is 65.8.

SOLUTIONS AND STANDARDS OF BEHAVIOR

43

Let us note that no such statement of principles is ever satisfactory
per se> and the arguments adduced in its favor are usually either those of
inner stability or of less clearly defined kinds of desirability, mainly con-
cerning distribution.

Little can be said about the latter type of motivation. Our problem
is not to determine what ought to happen in pursuance of any set of —
necessarily arbitrary — a priori principles, but to investigate where the
equilibrium of forces lies.

As far as the first motivation is concerned, it has been our aim to give
just those arguments precise and satisfactory form, concerning both global
aims and individual apportionments. This made it necessary to take up
the entire question of inner stability as a problem in its own right. A theory
which is consistent at this point cannot fail to give a precise account of the
entire interplay of economic interests, influence and power.

4.7. Games and Social Organizations

4 . 7 . It may now be opportune to revive the analogy with games, which
we purposely suppressed in the previous paragraphs (cf. footnote 1 on
p. 41). The parallelism between the solutions S in the sense of 4.5.3. on
one hand and of stable “ standards of behavior” on the other can be used
for corroboration of assertions concerning these concepts in both directions.
At least we hope that this suggestion will have some appeal to the reader.
We think that the procedure of the mathematical theory of games of
strategy gains definitely in plausibility by the correspondence which exists
between its concepts and those of social organizations. On the other
hand, almost every statement which we — or for that matter anyone else —
ever made concerning social organizations, runs afoul of some existing
opinion. And, by the very nature of things, most opinions thus far could
hardly have been proved or disproved within the field of social theory.
It is therefore a great help that all our assertions can be borne out by specific
examples from the theory of games of strategy.

Such is indeed one of the standard techniques of using models in the
physical sciences. This two-way procedure brings out a significant func-
tion of models, not emphasized in their discussion in 4.1.3.

To give an illustration: The question whether several stable “ orders
of society” or “ standards of behavior” based on the same physical back-
ground are possible or not, is highly controversial. There is little hope
that it will be settled by the usual methods because of the enormous com-
plexity of this problem among other reasons. But we shall give specific
examples of games of three or four persons, where one game possesses several
solutions in the sense of 4.5.3. And some of these examples will be seen
to be models for certain simple economic problems. (Cf. 62.)

4.8. Concluding Remarks

4 . 8 . 1 . In conclusion it remains to make a few remarks of a more formal
nature.

44

FORMULATION OF THE ECONOMIC PROBLEM

We begin with this observation: Our considerations started with single
imputations — which were originally quantitative extracts from more
detailed combinatorial sets of rules. From these we had to proceed to
sets S of imputations, which under certain conditions appeared as solutions.
Since the solutions do not seem to be necessarily unique, the complete
answer to any specific problem consists not in finding a solution, but in
determining the set of all solutions. Thus the entity for which we look in
any particular problem is really a set of sets of imputations. This may seem
to be unnaturally complicated in itself ; besides there appears no guarantee
that this process will not have to be carried further, conceivably because
of later difficulties. Concerning these doubts it suffices to say: First, the
mathematical structure of the theory of games of strategy provides a formal
justification of our procedure. Second, the previously discussed connections
with “standards of behavior” (corresponding to sets of imputations) and
of the multiplicity of “standards of behavior” on the same physical back-
ground (corresponding to sets of sets of imputations) make just this amount
of complicatedness desirable.

One may criticize our interpretation of sets of imputations as “standards
of behavior.” Previously in 4.1.2. and 4.1.4. we introduced a more ele-
mentary concept, which may strike the reader as a direct formulation of a
“standard of behavior”: this was the preliminary combinatorial concept
of a solution as a set of rules for each participant, telling him how to behave
in every possible situation of the game. (From these rules the single
imputations were then extracted as a quantitative summary, cf. above.)
Such a simple view of the “standard of behavior” could be maintained,
however, only in games in which coalitions and the compensations between
coalition partners (cf. 4.3.2.) play no role, since the above rules do not
provide for these possibilities. Games exist in which coalitions and compen-
sations can be disregarded: e.g. the two-person game of zero-sum mentioned
in 4.2.3., and more generally the “inessential” games to be discussed in
27.3. and in (31 :P) of 31.2.3. But the general, typical game — in particular
all.significant problems of a social exchange economy— cannot be treated with-
out these devices. Thus the same arguments which forced us to consider sets
of imputations instead of single imputations necessitate the abandonment
of that narrow concept of “standard of behavior.” Actually we shall call
these sets of rules the “strategies” of the game.

4 . 8 . 2 . The next subject to be mentioned concerns the static or dynamic
nature of the theory. We repeat most emphatically that our theory is
thoroughly static. A dynamic theory would unquestionably be more
complete and therefore preferable. But there is ample evidence from other
branches of science that it is futile to try to build one as long as the static
side is not thoroughly understood. On the other hand, the reader may
object to some definitely dynamic arguments which were made in the course
of our discussions. This applies particularly to all considerations concern-
ing the interplay of various imputations under the influence of “domina-

SOLUTIONS AND STANDARDS OF BEHAVIOR 45

tion,” cf. 4.6.2. We think that this is perfectly legitimate. A static
theory deals with equilibria. 1 The essential characteristic of an equilibrium
is that it has no tendency to change, i.e. that it is not conducive to dynamic
developments. An analysis of this feature is, of course, inconceivable
without the use of certain rudimentary dynamic concepts. The important
point is that they are rudimentary. In other words: For the real dynamics
which investigates the precise motions, usually far away from equilibria, a
much deeper knowledge of these dynamic phenomena is required. 2 * 3

4 . 8 . 3 . Finally let us note a point at which the theory of social phenomena
will presumably take a very definite turn away from the existing patterns of
mathematical physics. This is, of course, only a surmise on a subject where
much uncertainty and obscurity prevail.

Our static theory specifies equilibria — i.e. solutions in the sense of 4.5.3.
— which are sets of imputations. A dynamic theory — when one is found —
will probably describe the changes in terms of simpler concepts: of a single
imputation — valid at the moment under consideration — or something
similar. This indicates that the formal structure of this part of the theory —
the relationship between statics and dynamics — may be generically different
from that of the classical physical theories. 4

All these considerations illustrate once more what a complexity of
theoretical forms must be expected in social theory. Our static analysis
alone necessitated the creation of a conceptual and formal mechanism which
is very different from anything used, for instance, in mathematical physics.
Thus the conventional view of a solution as a uniquely defined number or
aggregate of numbers was seen to be too narrow for our purposes, in spite
of its success in other fields. The emphasis on mathematical methods
seems to be shifted more towards combinatorics and set theory — and away
from the algorithm of differential equations which dominate mathematical
physics.

1 The dynamic theory deals also with inequilibria — even if they are sometimes called
“dynamic equilibria.”

1 The above discussion of statics versus dynamics is, of course, not at all a construction
ad hoc. The reader who is familiar with mechanics for instance will recognize in it a
reformulation of well known features of the classical mechanical theory of statics and
dynamics. What we do claim at this time is that this is a general characteristic of
scientific procedure involving forces and changes in structures.

* The dynamic concepts which enter into the discussion of static equilibria are parallel
to the “virtual displacements ” in classical mechanics. The reader may also remember at
this point the remarks about “virtual existence” in 4.3.3.

4 Particularly from classical mechanics. The analogies of the type used in footnote 2
above, cease at this point.

CHAPTER II

GENERAL FORMAL DESCRIPTION OF GAMES OF STRATEGY

5. Introduction

5.1. Shift of Emphasis from Economics to Games

6 . 1 . It should be clear from the discussions of Chapter I that a theory
of rational behavior — i.e. of the foundations of economics and of the main
mechanisms of social organization — requires a thorough study of the “games
of strategy.” Consequently we must now take up the theory of games as an
independent subject. In studying it as a problem in its own right, our
point of view must of necessity undergo a serious shift. In Chapter I our
primary interest lay in economics. It was only after having convinced
ourselves of the impossibility of making progress in that field without a
previous fundamental understanding of the games that we gradually
approached the formulations and the questions which are partial to that
subject. But the economic viewpoints remained nevertheless the dominant
ones in all of Chapter I. From this Chapter II on, however, we shall have
to treat the games as games. Therefore we shall not mind if some points
taken up have no economic connections whatever, — it would not be possible
to do full justice to the subject otherwise. Of course most of the main
concepts are still those familiar from the discussions of economic literature
(cf. the next section) but the details will often be altogether alien to it —
and details, as usual, may dominate the exposition and overshadow the
guiding principles.

6.2. General Principles of Classification and of Procedure

6 . 2 . 1 . Certain aspects of “games of strategy” which were already
prominent in the last sections of Chapter I will not appear in the beginning
stages of the discussions which we are now undertaking. Specifically:
There will be at first no mention of coalitions between players and the
compensations which they pay to each other. (Concerning these, cf.
4.3.2., 4.3.3., in Chapter I.) We give a brief account of the reasons, which
will also throw some light on our general disposition of the subject.

An important viewpoint in classifying games is this: Is the sum of all
payments received by all players (at the end of the game) always zero; or
is this not the case? If it is zero, then one can say that the players pay only
to each other, and that no production or destruction of goods is involved.
All games which are actually played for entertainment are of this type. But
the economically significant schemes are most essentially not such. There
the sum of all payments, the total social product, will in general not be

.46

INTRODUCTION

47

zero, and not even constant. I.e., it will depend on the behavior of the
players — the participants in the social economy. This distinction was
already mentioned in 4.2.1., particularly in footnote 2, p. 34. We shall call
games of the first-mentioned type zero-sum games, and those of the latter
type non-zero-sum games.

We shall primarily construct a theory of the zero-sum games, but it will
be found possible to dispose, with its help, of all games, without restriction.
Precisely: We shall show that the general (hence in particular the variable
sum) n - person game can be reduced to a zero-sum n + I-person game.
(Cf. 56.2.2.) Now the theory of the zero-sum n - person game will be based
on the special case of the zero-sum two-person game. (Cf. 25.2.) Hence
our discussions will begin with a theory of these games, which will indeed
be carried out in Chapter III.

Now in zero-sum two-person games coalitions and compensations
can play no role. 1 The questions which are essential in these games are
of a different nature. These are the main problems: How does each
player plan his course — i.e. how does one formulate an exact concept of a
strategy? What information is available to each player at every stage
of the game? What is the role of a player being informed about the other
player’s strategy? About the entire theory of the game?

5 . 2 . 2 . All these questions are of course essential in all games, for any
number of players, even when coalitions and compensations have come into
their own. But for zero-sum two-person games they are the only ones
which matter, as our subsequent discussions will show. Again, all these
questions have been recognized as important in economics, but we think that
in the theory of games they arise in a more elementary — as distinguished
from composite — fashion. They can, therefore, be discussed in a precise
way and — as we hope to show — be disposed of. But in the process of this
analysis it will be technically advantageous to rely on pictures and examples
which are rather remote from the field of economics proper, and belong
strictly to the field of games of the conventional variety. Thus the dis-
cussions which follow will be dominated by illustrations from Chess,
“ Matching Pennies,” Poker, Bridge, etc., and not from the structure of
cartels, markets, oligopolies, etc.

At this point it is also opportune to recall that we consider all trans-
actions at the end of a game as purely monetary ones — i.e. that we ascribe
to all players an exclusively monetary profit motive. The meaning of this
in terms of the utility concept was analyzed in 2.1.1. in Chapter I. For the
present — particularly for the “ zero-sum two-person games” to be discussed

1 The only fully satisfactory “proof” of this assertion lies in the construction of a
complete theory of all zero-sum two-person games, without use of those devices. This
will be done in Chapter III, the decisive result being contained in 17. It ought to be
clear by common sense, however, that “understandings” and “coalitions” can have no
role here: Any such arrangement must involve at least two players — hence in this case all
players — for whom the sum of payments is identically zero. I.e. th re are no opponents
left and no possible objectives.

48

DESCRIPTION OF GAMES OF STRATEGY

first (cf. the discussion of 5.2.1.) — it is an absolutely necessary simplifi-
cation. Indeed, we shall maintain it through most of the theory, although
variants will be examined later on. (Cf. Chapter XII, in particular 66.)

6 . 2 . 3 . Our first task is to give an exact definition of what constitutes a
game. As long as the concept of a game has not been described with
absolute mathematical — combinatorial — precision, we cannot hope to
give exact and exhaustive answers to the questions formulated at the end
of 5.2.1. Now while our first objective is — as was explained in 5.2.1. — the
theory of zero-sum two-person games, it is apparent that the exact descrip-
tion of what constitutes a game need not be restricted to this case. Conse-
quently we can begin with the description of the general n-person game.
In giving this description we shall endeavor to do justice to all conceivable
nuances and complications which can arise in a game — insofar as they are
not of an obviously inessential character. In this way we reach — in several
successive steps — a rather complicated but exhaustive and mathematically
precise scheme. And then we shall see that it is possible to replace this
general scheme by a vastly simpler one, which is nevertheless fully and
rigorously equivalent to it. Besides, the mathematical device which
permits this simplification is also of an immediate significance for our
problem: It is the introduction of the exact concept of a strategy.

It should be understood that the detour — which leads to the ultimate,
simple formulation of the problem, over considerably more complicated
ones — is not avoidable. It is necessary to show first that all possible
complications have been taken into consideration, and that the mathe-
matical device in question does guarantee the equivalence of the involved
setup to the simple.

All this can — and must — be done for all games, of any number of play-
ers. But after this aim has been achieved in entire generality, the next
objective of the theory is — as mentioned above — to find a complete solution
for the zero-sum two-person game. Accordingly, this chapter will deal
with all games, but the next one with zero-sum two-person games only. After
they are disposed of and some important examples have been discussed, we
shall begin to re-extend the scope of the investigation — first to zero-sum n-
person games, and then to all games.

Coalitions and compensations will only reappear during this latter stage.

6. The Simplified Concept of a Game
6.1. Explanation of the Termini Technici

6 . 1 . Before an exact definition of the combinatorial concept of a game
can be given, we must first clarify the use of some termini. There are
some notions which are quite fundamental for the discussion of games,
but the use of which in everyday language is highly ambiguous. The words
which describe them are used sometimes in one sense, sometimes in another,
and occasionally — worst of all — as if they were synonyms. We must

SIMPLIFIED CONCEPT OF A GAME

49

therefore introduce a definite usage of termini technici ) and rigidly adhere
to it in all that follows.

First, one must distinguish between the abstract concept of a game,
and the individual plays of that game. The game is simply the totality
of the rules which describe it. Every particular instance at which the
game is played — in a particular way — from beginning to end, is a play. 1

Second, the corresponding distinction should be made for the moves,
which are the component elements of the game. A move is the occasion
of a choice between various alternatives, to be made either by one of the
players, or by some device subject to chance, under conditions precisely
prescribed by the rules of the game. The move is nothing but this abstract
“occasion,” with the attendant details of description, — i.e. a component
of the game. The specific alternative chosen in a concrete instance — i.e.
in a concrete play — is the choice. Thus the moves are related to the
choices in the same way as the game is to the play. The game consists
of a sequence of moves, and the play of a sequence of choices. 2

Finally, the rules of the game should not be confused with the strategies
of the players. Exact definitions will be given subsequently, but the
distinction which we stress must be clear from the start. Each player
selects his strategy — i.e. the general principles governing his choices — freely.
While any particular strategy may be good or bad — provided that these
concepts can be interpreted in an exact sense (cf. 14.5. and 17.8-17.10.) —
it is within the players discretion to use or to reject it. The rules of the
game, however, are absolute commands. If they are ever infringed, then
the whole transaction by definition ceases to be the game described by those
rules. In many cases it is even physically impossible to violate them. 3

6.2. The Elements of the Game

6.2.1. Let us now consider a game T of n players who, for the sake of
brevity, will be denoted by 1, • • • , n. The conventional picture provides
that this game is a sequence of moves, and we assume that both the number
and the arrangement of these moves is given ab initio. We shall see later
that these restrictions are not really significant, and that they can be
removed without difficulty. For the present let us denote the (fixed)
number of moves in T by v — this is an integer v = 1, 2, • • • . The moves
themselves we denote by 9fTli, • • • , 9Tl„ and we assume that this is the
chronological order in which they are prescribed to take place.

1 In most games everyday usage calls a play equally a game ; thus in chess, in poker,
in many sports, etc. In Bridge a play corresponds to a “rubber,” in Tennis to a “set,”
but unluckily in these games certain components of the play are again called “games.”
The French terihinology is tolerably unambiguous: “game” * “jeu,” “play” =*
“partie.”

1 In this sense we would talk in chess of the first move, and of the choice “E2-E4.”

• E.g. : In Chess the rules of the game forbid a player to move his king into a position
of “check.” This is a prohibition in the same absolute sense in which he may not move a
pawn sideways. But to move the king into a position where the opponent can “check-
mate” him at the next move is merely unwise, but not forbidden.

50

DESCRIPTION OF GAMES OF STRATEGY

Every move 911*, k = 1, • • • , v , actually consists of a number of
alternatives, among which the choice — which constitutes the move 911* —
takes place. Denote the number of these alternatives by a K and the
alternatives themselves by Ct*(l), • • * , Ot*(a*).

The moves are of two kinds. A move of the first kind , or a personal
move f is a choice made by a specific player, i.e. depending on his free decision
and nothing else. A move of the second kind, or a chance move, is a choice
depending on some mechanical device, which makes its outcome fortuitous
with definite probabilities. 1 Thus for every personal move it must be
specified which player’s decision determines this move, whose move it is.
We denote the player in question (i.e. his number) by fc*. So k K = 1, • • • ,
n. For a chance move we put (conventionally) k K = 0. In this case the
probabilities of the various alternatives a*(l), • • • , d K (a K ) must be given.
We denote these probabilities by p*(l), • • • , p K (a K ) respectively. 2

6.2.2. In a move 911* the choice consists of selecting an alternative
G«(l), ' ’ * > G«(a«)> i.e. its number 1, • • • , a*. We denote the number
so chosen by <r*. Thus this choice is characterized by a number cr* = 1,
And the complete play is described by specifying all choices,
corresponding to all moves 91li, • • • , 91Z„. I.e. it is described by a sequence
Ci, , c p .

Now the rule of the game T must specify what the outcome of the play
is for each player k = 1, • • • , n, if the play is described by a given sequence
ci, * - - a,. I.e. what payments every player receives when the play is
completed. Denote the payment to the player k by $ k ($k > 0 if k receives
a payment, < 0 if he must make one, $ k = 0 if neither is the case).
Thus each 5 k must be given as a function of the ci, • • • , c ¥ :

5 k = $ k (c i, • • • , c v ), k = 1, • • • , n.

We emphasize again that the rules of the game T specify the function
• * * , c p ) only as a function, 3 i.e. the abstract dependence of each

on the variables c h • • • , c ¥ . But all the time each c K is a variable,
with the domain of variability 1, • • • , a K . A specification of particular
numerical values for the c K , i.e. the selection of a particular sequence c h
* * * > is no part of the game T. It is, as we pointed out above, the
definition of a play.

1 E.g., dealing cards from an appropriately shuffled deck, throwing dice, etc. It is
even possible to include certain games of strength and skill, where “strategy ” plays a role,
e.g. Tennis, Football, etc. In these the actions of the players are up to a certain point
personal moves — i.e. dependent upon their free decision — and beyond this point chance
moves, the probabilities being characteristics of the player in question.

‘Since the p«(l), • • • , p«(a«) are probabilities, they are necessarily numbers 0.
Since they belong to disjunct but exhaustive alternatives, their sum (for a fixed k) must
be one. I.e. :

<*K

Pk(<t) 0, ^ p K (<r) « 1.

<r-l

3 For a systematic exposition of the conoept of a function cf. 13.1.

SIMPLIFIED CONCEPT OF A GAME 51

6.3. Information and Preliminarity

6 . 3 . 1 . Our description of the game T is not yet complete. We have
failed to include specifications about the state of information of every
player at each decision which he has to make, — i.e. whenever a personal
move turns up which is his move. Therefore we now turn to this aspect
of the matter.

This discussion is best conducted by following the moves • • • ,
as the corresponding choices are made.

Let us therefore fix our attention on a particular move If this
9TC« is a chance move, then nothing more need be said: the choice is decided
by chance; nobody’s will and nobody’s knowledge of other things can
influence it. But if 9TI, is a personal move, belonging to the player k K , then
it is quite important what fc,’s state of information is when he forms his
decision concerning 9TC* — i.e. his choice of <r K .

The only things he can be informed about are the choices corresponding
to the moves preceding 9fTl« — the moves SfTCi, • • • , I.e. he may know

the values of <n, • • • , ay-i. But he need not know that much. It is an
important peculiarity of T, just how much information concerning a i, • • • ,
cr*_i the player k K is granted, when he is called upon to choose <r K . We
shall soon show in several examples what the nature of such limitations is.

The simplest type of rule which describes fc*’s state of information at 9NI*
is this: a set A* consisting of some numbers from among X = 1, • • * , k — 1,
is given. It is specified that k K knows the values of the <r\ with X belong-
ing to A*, and that he is entirely ignorant of the a\ with any other X.

In this case we shall say, when X belongs to A„ that X is preliminary
to k. This implies X = 1, • • • , #c — 1, i.e. X < k, but need not be implied
by it. Or, if we consider, instead of X, k, the corresponding moves 9Hx, 3Tl«:
Preliminarity implies anteriority, 1 but need not be implied by it.

6 . 3 . 2 . In spite of its somewhat restrictive character, this concept of
preliminarity deserves a closer inspection. In itself, and in its relationship
to anteriority (cf. footnote 1 above), it gives occasion to various combina-
torial possibilities. These have definite meanings in those games in which
they occur, and we shall now illustrate them by some examples of particu-
larly characteristic instances.

6.4. Preliminarity, Transitivity, and Signaling

6 . 4 . 1 . We begin by observing that there exist games in which pre-
liminarity and anteriority are the same thing. I.e., where the players k K
who makes the (personal) move 9TL is informed about the outcome of the
choices of all anterior moves 3TCi, • • • , i. Chess is a typical representa-
tive of this class of games of “perfect” information. They are generally
considered to be of a particularly rational character. We shall see in 15.,
specifically in 15.7., how this can be interpreted in a precise way.

1 In time, X < k means that 9TC\ occurs before 9 Tl*.

52

DESCRIPTION OF GAMES OF STRATEGY

Chess has the further feature that all its moves are personal. Now it
is possible to conserve the first-mentioned property — the equivalence of
preliminarity and anteriority — even in games which contain chance moves.
Backgammon is an example of this. 1 Some doubt might be entertained
whether the presence of chance moves does not vitiate the “ rational char-
acter” of the game mentioned in connection with the preceding examples.

We shall see in 15.7.1. that this is not so if a very plausible interpretation
of that “rational character” is adhered to. It is not important whether
all moves are personal or not; the essential fact is that preliminarity and
anteriority coincide.

6.4.2. Let us now consider games where anteriority does not imply
preliminarity. I.e., where the player k K who makes the (personal) move 9TC,
is not informed about everything that happened previously. There is a
large family of games in which this occurs. These games usually con-
tain chance moves as well as personal moves. General opinion considers
them as being of a mixed character: while their outcome is definitely
dependent on chance, they are also strongly influenced by the strategic
abilities of the players.

Poker and Bridge are good examples. These two games show, further-
more, what peculiar features the notion of preliminarity can present,
once it has been separated from anteriority. This point perhaps deserves
a little more detailed consideration.

Anteriority, i.e. the chronological ordering of the moves, possesses
the property of transitivity. 2 Now in the present case, preliminarity
need not be transitive. Indeed it is neither in Poker nor in Bridge, and the
conditions under which this occurs are quite characteristic.

Poker: Let 9TC M be the deal of his “hand” to player 1 — a chance move;
3TCx the first bid of player 1 — a personal move of 1 ; 9TC, the first (subsequent)
bid of player 2 — a personal move of 2. Then is preliminary to 3Tlx and
9Tlx to but is not preliminary to 9TI,. 3 Thus we have intransitivity,
but it involves both players. Indeed, it may first seem unlikely that
preliminarity could in any game be intransitive among the personal moves
of one particular player. It would require that this player should “forget”
between the moves 9TCx and 9TC* the outcome of the choice connected with
9TC„ 4 — and it is difficult to see how this “forgetting” could be achieved, and

1 The chance moves in Backgammon are the dice throws which decide the total num-
ber of steps by which each player’s men may alternately advance. The personal moves
are the decisions by which each player partitions that total number of steps allotted to
him among his individual men. Also his decision to double the risk, and his alternative
to accept or to give up when the opponent doubles. At every move, however, the out-
come of the choices of all anterior moves are visible to all on the board.

* I.e. : If is anterior to 9Tlx and 9Tlx to 2fTl« then 9Tl M is anterior to Special situa-
tions where the presence or absence of transitivity was of importance, were analyzed in
4.4.2., 4.6.2. of Chapter I in connection with the relation of domination.

3 I.e., 1 makes his first bid knowing his own “hand”; 2 makes his first bid knowing
l’s (preceding) first bid; but at the same time 2 is ignorant of l's “hand.”

4 We assume that 9TC M is preliminary to SHlx and 91tx to 9Tl« but 9R M not to 9TI*.

SIMPLIFIED CONCEPT OF A GAME 53

even enforced I Nevertheless our next example provides an instance of
just this.

Bridge: Although Bridge is played by 4 persons, to be denoted by
A,B,C,D, it should be classified as a two-person game. Indeed, A and C
form a combination which is more than a voluntary coalition, and so do
B and D. For A to cooperate with B (or D ) instead of with C would be
“cheating,” in the same sense in which it would be “cheating” to look into
B’s cards or failing to follow suit during the play. I.e. it would be a viola-
tion of the rules of the game. If three (or more) persons play poker, then
it is perfectly permissible for two (or more) of them to cooperate against
another player when their interests are parallel — but in Bridge A and C
(and similarly B and D) must cooperate, while A and B are forbidden to
cooperate. The natural way to describe this consists in declaring that A
and C are really one player 1, and that B and D are really one player 2.
Or, equivalently: Bridge is a two-person game, but the two players 1 and 2
do not play it themselves. 1 acts through two representatives A and C and
2 through two representatives B and D.

Consider now the representatives of 1, A and C. The rules of the game
restrict communication, i.e. the exchange of information, between them.
E.g. : let 9TC m be the deal of his “hand” to A — a chance move; 9Tlx the first
card played by A — a personal move of 1 ; 911* the card played into this trick
by C — a personal move of 1. Then 9Tl M is preliminary to 91lx and 9Tlx to 9TI*
but 9Tl M is not preliminary to 911*. 1 Thus we have again intransitivity, but
this time it involves only one player. It is worth noting how the necessary
“forgetting” of 9TC M between 9fl x and 9TI* was achieved by “splitting the
personality” of 1 into A and C.

6.4.3. The above examples show that intransitivity of the relation of
preliminarity corresponds to a very well known component of practical
strategy: to the possibility of “signaling.” If no knowledge of 9Tl M is
available at 9TC«, but if it is possible to observe affix's outcome at 9Tl« and 9Tlx
has been influenced by 9Tl M (by knowledge about 9TI/S outcome), then
9llx is really a signal from 9TC M to 9TI* — a device which (indirectly) relays
information. Now two opposite situations develop, according to whether
9flx and 9TI* are moves of the same player, or of two different players.

In the first case — which, as we saw, occurs in Bridge — the interest of
the player (who is k\ = fc*). lies in promoting the “signaling,” i.e. the
spreading of information “within his own organization.” This desire
finds its realization in the elaborate system of “conventional signals” in
Bridge. 2 These are parts of the strategy, and not of the rules of the game

1 I.e. A plays his first card knowing his own “hand ”; C contributes to this trick know-
ing the (initiating) card played by A ; but at the same time C is ignorant of A * s “hand.”

* Observe that this “signaling” is considered to be perfectly fair in Bridge if it is
carried out by actions which are provided for by the rules of the game. E.g. it is correct
for A and C (the two components of player 1, cf. 6.4.2.) to agree — before the play begins!
— that an “original bid” of two trumps “indicates” a weakness of the other suits. But
it is incorrect — i.e. “cheating” — to indicate a weakness by an inflection of the voice at
bidding, or by tapping on the table, etc.

54

DESCRIPTION OF GAMES OF STRATEGY

(cf. 6.1.), and consequently they may vary, 1 while the game of Bridge
remains the same.

In the second case — which, as we saw, occurs in Poker — the interest
of the player (we now mean k\ , observe that here k\ k K ) lies in preventing
this “signaling,” i.e. the spreading of information to the opponent ( k K ).
This is usually achieved by irregular and seemingly illogical behavior
(when making the choice at 9fH\) — this makes it harder for the opponent
to draw inferences from the outcome of SJTlx (which he sees) concerning the
outcome of 9H M (of which he has no direct news). I.e. this procedure makes
the “signal” uncertain and ambiguous. We shall see in 19.2.1. that this is
indeed the function of “bluffing” in Poker. 2

We shall call these two procedures direct and inverted signaling. It ought
to be added that inverted signaling — i.e. misleading the opponent — occurs
in almost all games, including Bridge. This is so since it is based on the
intransitivity of preliminarity when several players are involved, which is
easy to achieve. Direct signaling, on the other hand, is rarer; e.g. Poker
contains no vestige of it. Indeed, as we pointed out before, it implies the
intransitivity of preliminarity when only one player is involved — i.e. it
requires a well-regulated “forgetfulness” of that player, which is obtained in
Bridge by the device of “splitting the player up ” into two persons.

At any rate Bridge and Poker seem to be reasonably characteristic
instances of these two kinds of intransitivity — of direct and of inverted
signaling, respectively.

Both kinds of signaling lead to a delicate problem of balancing in actual
playing, i.e. in the process of trying to define “good,” “rational” playing.
Any attempt to signal more or to signal less than “unsophisticated” playing
would involve, necessitates deviations from the “unsophisticated” way of
playing. And this is usually possible only at a definite cost, i.e. its direct
consequences are losses. Thus the problem is to adjust this “extra” signal-
ing so that its advantages — by forwarding or by withholding information —
overbalance the losses which it causes directly. One feels that this involves
something like the search for an optimum, although it is by no means clearly
defined. We shall see how the theory of the two-person game takes care
already of this problem, and we shall discuss it exhaustively in one charac-
teristic instance. (This is a simplified form of Poker. Cf. 19.)

Let us observe, finally, that all important examples of intransitive
preliminarity are games containing chance moves. This is peculiar, because
there is no apparent connection between these two phenomena. 3 - 4 Our

1 They may even be different for the two players, i.e. for A and C on one hand and
B and D on the other. But ‘‘within the organization” of one player, e.g. for A and C,
they must agree.

* And that “bluffing” is not at all an attempt to secure extra gains — in any direct
sense — when holding a weak hand. Cf. loc. cit.

* Cf. the corresponding question when preliminarity coincides with anteriority, and
thus is transitive, as discussed in 6.4.1. As mentioned there, the presence or absence of
chance moves is immaterial in that case.

4 “Matching pennies” is an example which has a certain importance in this connec-
tion. This and other related games will be discussed in 18.

(COMPLETE CONCEPT OF A GAME

55

subsequent analysis will indeed show that the presence or absence of chance
moves scarcely influences the essential aspects of the strategies in this
situation.

7. The Complete Concept of a Game
7.1. Variability of the Characteristics of Each Move

7.1.1. We introduced in 6.2.1. the a K alternatives G«(l), • • • , Gt«(a«)
of the move 9TI*. Also the index fc* which characterized the move as a
personal or chance one, and in the first case the player whose move it is;
and in the second case the probabilities p*(l), * * * , p K (a K ) of the above alter-
natives. We described in 6.3.1. the concept of preliminarity with the help
of the sets A*, — this being the set of all X (from among the X = 1 , • • • , k — 1)
which are preliminary to k. We failed to specify, however, whether all
these objects — a K} k K) A« and the Ot«(cr), p K (<r) for a = 1, • • • , a K — depend
solely on k or also on other things. These “other things” can, of course,
only be the outcome of the choices corresponding to the moves which are
anterior to 9Tl«. I.e. the numbers <n, • • • , <r,_ i. (Cf. 6.2.2.)

This dependence requires a more detailed discussion.

First, the dependence of the alternatives & K (<r) themselves (as distin-
guished from their number a K \) on a h • • • , is immaterial. We may
as well assume that the choice corresponding to the move 3TC* is made not
between the &«(<r) themselves, but between their numbers <r. In fine , it is
only the <r of 9fTC*, i.e. <r„ which occurs in the expressions describing the out-
come of the play, — i.e. in the functions SF*(< ti, • • • , a K ), k = 1, • • • , n. 1
(Cf. 6.2.2.)

Second, all dependences (on a x , • • • , <t,_i) which arise when 3m* turns
out to be a chance move — i.e. when k K = 0 (cf. the end of 6.2.1.) — cause no
complications. They do not interfere with our analysis of the behavior of
the players. This disposes, in particular, of all probabilities p,(<r), since
they occur only in connection with chance moves. (The A*, on the other
hand, never occur in chance moves.)

Third, we must consider the dependences (on a h • • • , <r«_i) of the
a K , k K , A* when 3TC* turns out to be a personal move. 2 Now this possibility
is indeed a source of complications. And it is a very real possibility. 3 The
reason is this.

1 The form and nature of the alternatives a K {<r) offered at might, of course, convey

to the player k K (if 3TC* is a personal move) some information concerning the anterior
<n, • • • , <r K - 1 values, — if the Ct,c(<r) depend on those. But any such information should
be specified separately, as information available to k K at 9lt*. We have discussed the
simplest schemes concerning the subject of information in 6.3.1., and shall complete the
discussion in 7.1.2. The discussion of a*, A«, which follows further below, is charac-

teristic also as far as the role of the Ot«(<r) as possible sources of information is concerned.

2 Whether this happens for a given *, will itself depend on k K — and hence indirectly
on <ti, • • • , <r«-i — since it is characterized by k K ^ 0 (cf. the end of 6.2.1.).

s E.g.: In Chess the number of possible alternatives a* at 9Tl« depends on the positions
of the men, i.e. the previous course of the play. In Bridge the player who plays the first

56

DESCRIPTION OF GAMES OF STRATEGY

7.1.2. The player k K must be informed at SHI* of the values of a K ,

A* — since these are now part of the rules of the game which he must observe.
Insofar as they depend upon a h • • •, cr,_i, he may draw from them certain
conclusions concerning the values of a h • • • , But he is supposed

to know absolutely nothing concerning the <r\ with X not in A*! It is hard
to see how conflicts can be avoided.

To be precise: There is no conflict in this special case: Let A, be inde-
pendent of all <r i, • • • , 0 v_i, and let a K , k K depend only on the <x\ with X in A*.
Then the player k K can certainly not get any information from ct K) k K) A«
beyond what he knows anyhow (i.e. the values of the <r\ with X in A«). If
this is the case, we say that we have the special form of dependence.

But do we always have the special form of dependence? To take an
extreme case: What if A* is always empty — i.e. k K expected to be completely
uninformed at 91X< — and yet e.g. a K explicitly dependent on some of the
<r i, • • * , <7«_l!

This is clearly inadmissible. We must demand that all numerical con-
clusions which can be derived from the knowledge of a K , k K) A„ must be
explicitly and ab initio specified as information available to the player k K
at 9Tl«. It would be erroneous, however, to try to achieve this by including
in A* the indices X of all these ox, on which a K , k K , A* explicitly depend. In
the first place great care must be exercised in order to avoid circularity in
this requirement, as far as A* is concerned. 1 But even if this difficulty does
not arise, because A* depends only on k and not on <T h • • • , ow — i.e. if the
information available to every player at every moment is independent of
the previous course of the play — the above procedure may still be inadmis-
sible. Assume, e.g., that a K depends on a certain combination of some a\
from among the X = 1, ••*,* — 1, and that the rules of the game do
indeed provide that the player k K at 3TC* should know the value of this com-
bination, but that it does not allow him to know more (i.e. the values of the
individual a i, • • • , <r*_i). E.g.: He may know the value of + <r\ where

/x, X are both anterior to k (//, X < x), but he is not allowed to know the
separate values of <r M and a\.

One could try various tricks to bring back the above situation to our
earlier, simpler, scheme, which describes k K y s state of information by means
of the set A,. 2 But it becomes completely impossible to disentangle the
various components of k K y s information at 9TC*, if they themselves originate
from personal moves of different players, or of the same player but in

card to the next trick, i.e. k K at 9TI*, is the one who took the last trick, i.e. again dependent
upon the previous course of the play. In some forms of Poker, and some other related
games, the amount of information available to a player at a given moment, i.e. A* at 9^1*,
depends on what he and the others did previously.

1 The a\ on which, among others, A* depend are only defined if the totality of all A*,
for all sequences a, • • • , is considered. Should every A* contain these X?

2 In the above example one might try to replace the move by a new one in which
not a? is chosen, but -f <r\. SHI* would remain unchanged. Then k * at 9TC* would be
informed about the outcome of the choice connected with the new 9Tlu onlv.

COMPLETE CONCEPT OF A GAME

57

different stages of information. In our above example this happens if
fc M ^ K or ^ = h but the state of information of this player is not the

same at TO„ and at TOx. 1

7.2. The General Description

7.2.1. There are still various, more or less artificial, tricks by which one
could try to circumvent these difficulties. But the most natural procedure
seems to be to admit them, and to modify our definitions accordingly.

This is done by sacrificing the A* as a means of describing the state of
information. Instead, we describe the state of information of the player k K
at the time of his personal move TO* explicitly : By enumerating those func-
tions of the variable <r\ anterior to this move — i.e. of the <r h • • • , <r*_i — the
numerical values of which he is supposed to know at this moment. This is
a system of functions, to be denoted by <f>*.

So $* is a set of functions

h(a i, * * • , <r*_i).

Since the elements of <£* describe the dependence on <r l9 • • • , <r*_i, so $ K itself
is fixed, i.e. depending on k only. 2 «*, k K may depend on <r h • • • , <r*_ i, and
since their values are known to k K at 911*, these functions

a K = a«(cri, • • • , <r*_i), k K = k K {a i, • • • , <r*_i)

must belong to 3>*. Of course, whenever it turns out that k K = 0 (for a
special set of <n, • • • , (r*_i values), then the move TO* is a chance one (cf.
above), and no use will be made of <f>* — but this does not matter.

Our previous mode of description, with the A*, is obviously a special
case of the present one, with the $*. 3

7.2.2. At this point the reader may feel a certain dissatisfaction about
the turn which the discussion has taken. It is true that the discussion was
deflected into this direction by complications which arose in actual and
typical games (cf. footnote 3 on p. 55). But the necessity of replacing
the A* by the <$* originated in our desire to maintain absolute formal
(mathematical) generality. These decisive difficulties, which caused us
to take this step (discussed in 7.1.2., particularly as illustrated by the
footnotes there) were really extrapolated. I.e. they were not characteristic

1 In the instance of footnote 2 on p. 56, this means : If kp^ k \ , there is no player to whom
the new move (where + <r\ is chosen, and which ought to be personal) can be
attributed. If = k\ but the state of information varies from 9Tl M to TOx, then no state
of information can be satisfactorily prescribed for the new move TO*.

* This arrangement includes nevertheless the possibility that the state of information
expressed by $* depends on <n, • • • , <r«_ i. This is the case if, e.g., all functions
h(a i, • • • , <r«_i) of show an explicit dependence on o> for one set of values of <r\ , while
being independent of <r M for other values of <rx. Yet 4>* is fixed.

* If happens to consist of all functions of certain variables <rx — say of those for
which X belongs to a given set M, — and of no others, then the <£* description specializes
back to the A* one: A* being the above set M*. But we have seen that we cannot, in
general, count upon the existence of such a set.

58

DESCRIPTION OF GAMES OF STRATEGY

of the original examples, which are actual games. (E.g. Chess and Bridge
can be described with the help of the A<.)

There exist games which require discussion by means of the $«. But in
most of them one could revert to the A* by means of various extraneous
tricks — and the entire subject requires a rather delicate analysis upon which
it does not seem worth while to enter here. 1 There exist unquestionably
economic models where the $>« are necessary. 2

The most important point, however, is this.

In pursuit of the objectives which we have set ourselves we must achieve
the certainty of having exhausted all combinatorial possibilities in connec-
tion with the entire interplay of the various decisions of the players, their
changing states of information, etc. These are problems, which have been
dwelt upon extensively in economic literature. We hope to show that they
can be disposed of completely. But for this reason we want to be safe
against any possible accusation of having overlooked some essential possi-
bility by undue specialization.

Besides, it will be seen that all the formal elements which we are intro-
ducing now into the discussion do not complicate it ultima analysi. I.e.
they complicate only the present, preliminary stage of formal descrip-
tion. The final form of the problem turns out to be unaffected by them.
(Cf. 11.2.)

7 . 2 . 3 . There remains only one more point to discuss: The specializing
assumption formulated at the very start of this discussion (at the beginning
of 6.2.1.) that both the number and the arrangement of the moves are given
(i.e. fixed) ab initio. We shall now see that this restriction is not essential.

Consider first the “arrangement” of the moves. The possible varia-
bility of the nature of each move — i.e. of its k K — has already received full
consideration (especially in 7.2.1.). The ordering of the moves 9(TC«, k = 1,
• • • , v, was from the start simply the chronological one. Thus there is
nothing left to discuss on this score.

Consider next the number of moves v. This quantity too could be
variable, i.e. dependent upon the course of the play. 3 In describing this
variability of v a certain amount of care must be exercised.

1 We mean card games where players may discard some cards without uncovering
them, and are allowed to take up or otherwise use openly a part of their discards later.
There exists also a game of double-blind Chess — sometimes called “ Kriegsspiel " — which
belongs in this class. (For its description cf. 9.2.3. With reference to that description:
Each player knows about the ‘'possibility ' ’ of the other's anterior choices, without
knowing those choices themselves — and this “possibility" is a function of all anterior
choices.)

* Let a participant be ignorant of the full details of the previous actions of the others,
but let him be informed concerning certain statistical resultants of those actions.

1 It is, too, in most games: Chess, Backgammon, Poker, Bridge. In the case of Bridge
this variability is due first to the variable length of the “bidding" phase, and second to
the changing number of contracts needed to make a “rubber" (i.e. a play). Examples
of games with a fixed v are harder to find : we shall see that we can make v fixed in every
game by an artifice, but games in which v is ab initio fixed are apt to be monotonous.

COMPLETE CONCEPT OF A GAME

59

The course of the play is characterized by the sequence (of choices)
<ti, * • • , <r„ (cf. 6.2.2.). Now one cannot state simply that v may be a
function of the variables <r h • • • , because the full sequence <r if • • • ,
cannot be visualized at all, without knowing beforehand what its length v
is going to be. 1 The correct formulation is this: Imagine that the variables
<ri, 0 * 2 , crj, ■ • • are chosen one after the other. 2 If this succession of choices
is carried on indefinitely, then the rules of the game must at some place v
stop the procedure. Then v for which the stop occurs will, of course, depend
on all the choices up to that moment. It is the number of moves in that
particular play.

Now this stop rule must be such as to give a certainty that every con-
ceivable play will be stopped sometime. I.e. it must be impossible to
arrange the successive choices of a h <r 2 , <r 8 , • • • in such a manner (subject
to the restrictions of footnote 2 above) that the stop never comes. The
obvious way to guarantee this is to devise a stop rule for which it is
certain that the stop will come before a fixed moment, say v*. I.e. that
while v may depend on <r 2 , <73, • • • , it is sure to be v ^ v* where v*
does not depend on <r h <t 2 , cr 3 , • • • . If this is the case we say that the
stop rule is bounded by v*. We shall assume for the games which we con-
sider that they have stop rules bounded by (suitable, but fixed) numbers

„* 3,4

1 I.e. one cannot say that the length of the game depends on all choices made in con-
nection with all moves, since it will depend on the length of the game whether certain
moves will occur at all. The argument is clearly circular.

1 The domain of variability of <rj is 1, • • • , on. The domain of variability of a 2 is
' 1, • • • , and may depend on <j\ : a 2 = a 2 (<n). The domain of variability of <r» is
1, • • • , aj, and may depend on <r 1 , cr 2 : a a = crs(o’i, <r*) . Etc., etc.

3 This stop rule is indeed an essential part of every game. In most games it is easy
to find v’s fixed upper bound p*. Sometimes, however, the conventional form of the
rules of the game does not exclude that the play might — under exceptional conditions — go
on ad infinitum. In all these cases practical safeguards have been subsequently incor-
porated into the rules of the game with the purpose of securing the existence of the
bound p*. It must be said, however, that these safeguards are not always absolutely
effective — although the intention is clear in every instance, and even where exceptional
infinite plays exist they are of little practical importance. It is nevertheless quite
instructive, at least from a mathematical point of view, to discuss a few typical examples.

We give four examples, arranged according to decreasing effectiveness.

£cart6: A play is a “rubber," a “rubber" consists of winning two “games" out of
three (cf. footnote 1 on p. 49), a “game" consists of winning five “points,” and each
“deal" gives one player or the other one or two points. Hence a “rubber" is complete
after at most three “games," a “game" after at most nine “deals," and it is easy to
verify that a “deal" consists of 13, 14 or 18 moves. Hence v* = 3 ^9 • 18 — 486.

Poker: A priori two players could keep “overbidding" each other ad infinitum. It is
therefore customaiy to add to the rules a proviso limiting the permissible number of
“overbids." (The amounts of the bids are also limited, so as to make the number of
alternatives a K at these personal moves finite.) This of course secures a finite p *.

Bridge: The play is a “rubber" and this could go on forever if both sides (players)
invariably failed to make their contract. It is not inconceivable that the side which is in
danger of losing the “rubber," should in this way permanently prevent a completion of
the play by absurdly high bids. This is not done in practice, but there is nothing explicit
in the rules of the game to prevent it. In theory, at any rate, some stop rule should be
introduced in Bridge.

Chess: It is easy to construct sequences of choices (in the usual terminology:

60

DESCRIPTION OF GAMES OF STRATEGY

Now we can make use of this bound v* to get entirely rid of the variabil-
ity of v.

This is done simply by extending the scheme of the game so that there
are always v* moves SfTli, • • • , 9TI,*. For every sequence <r h <r*, <r 8 , • • •
everything is unchanged up to the move and all moves beyond £fTl„ are
“ dummy moves.” I.e. if we consider a move 3TC„ k = 1, • • • , v*, for a
sequence cn , era, <r 8 , • • • for which v < k, then we make 9TC, a chance move
with one alternative only 1 — i.e. one at which nothing happens.

Thus the assumptions made at the beginning of 6.2.1. — particularly
that v is given ab initio — are justified ex post.

8. Sets and Partitions

8.1. Desirability of a Set-theoretical Description of a Game

8.1. We have obtained a satisfactory and general description of the
concept of a game, which could now be restated with axiomatic precision
and rigidity to serve as a basis for the subsequent mathematical discussion.
It is worth while, however, before doing that, to pass to a different formula-
tion. This formulation is exactly equivalent to the one which we reached
in the preceding sections, but it is more unified, simpler when stated in a
general form, and it leads to more elegant and transparent notations.

In order to arrive at this formulation we must use the symbolism of
the theory of sets — and more particularly of partitions — more extensively
than we have done so far. This necessitates a certain amount of explana-
tion and illustration, which we now proceed to give.

“moves”) — particularly in the “end game” — which can go on ad infinitum without ever
ending the play (i.e. producing a “checkmate”). The simplest ones are periodical, i.e.
indefinite repetitions of the same cycle of choices, but there exist non-periodical ones as
well. All of them offer a very real possibility for the player who is in danger of losing to
secure sometimes a “ tie.” For this reason various “tie rules ” — i.e. stop rules — are in use
just to prevent that phenomenon.

One well known “tie rule” is this: Any cycle of choices (i.e. “moves”), when three
times repeated, terminates the play by a “tie.” This rule excludes most but not all
infinite sequences, and hence is really not effective.

Another “tie rule” is this: If no pawn has been moved and no officer taken (these
are “irreversible” operations, which cannot be undone subsequently) for 40 moves, then
the play is terminated by a “tie.” It is easy to see that this rule is effective, although the
v* is enormous.

4 From a purely mathematical point of view, the following question could be asked :
Let the stop rule be effective in this sense only, that it is impossible so to arrange the
successive choices <ri, <r*, <r a , • • • that the stop never comes. I.e. let there always be a
finite v dependent upon <ri, <rj, <x h • • • . Does this by itself secure the existence of a
fixed, finite v * bounding the stop rule? I.e. such that all v ^ p*?

The question is highly academic since all practical game rules aim to establish a p*
directly. (Cf., however, footnote 3 above.) It is nevertheless quite interesting
mathematically.

The answer is “ Yes,” i.e. p* always exists. Cf. e.g. D. K&nig: t)ber eine Schluss-
weise aus dem Endlichen ins Unendliche, Acta Litt. ac Scient. Univ. Szeged, Sect. Math.
Vol. III/II (1927) pp. 121-130; particularly the Appendix, pp. 129-130.

1 This means, of course, that = 1, K = 0, and p*(l) = 1.

SETS AND PARTITIONS

61

8.2. Sets, Their Properties, and Their Graphical Representation

8 . 2 . 1 . A set is an arbitrary collection of objects, absolutely no restriction
being placed on the nature and number of these objects, the elements
of the set in question. The elements constitute and determine the set as
such, without any ordering or relationship of any kind between them. I.e.
if two sets A, B are such that every element of A is also one of B and vice
versa , then they are identical in every respect, A = B. The relationship
of a being an element of the set A is also expressed by saying that a belongs
to A A

We shall be interested chiefly, although not always, in finite sets only, —
i.e. sets consisting of a finite number of elements.

Given any objects a, 0, 7, * • • we denote the set of which they are the
elements by (a, /3, 7, • • • ). It is also convenient to introduce a set which
contains no elements at all, the empty set . 2 We denote the empty set by ©.
We can, in particular, form sets with precisely one element, one-element sets.
The one-element set (a), and its unique element a, are not the same thing
and should never be confused. 3

We re-emphasize that any objects can be elements of a set. Of course
we shall restrict ourselves to mathematical objects. But the elements
can, for instance, perfectly well be sets themselves (cf. footnote 3), — thus
leading to sets of sets, etc. These latter are sometimes called by some other
— equivalent — name, e.g. systems or aggregates of sets. But this is not
necessary.

8 . 2 . 2 . The main concepts and operations connected with sets are these:

(8:A:a) A is a subset of B, or B a superset of A, if every element of

A is also an element of B. In symbols: A cBorBs A. A is
a proper subset of B } or B a proper superset of A, if the above is
true, but if B contains elements which are not elements of A.
In symbols: A c B or B ^ A. We see: If A is a subset of B and
B is a subset of A , then A = B. (This is a restatement of the
principle formulated at the beginning of 8.2.1.) Also: A is a
proper subset of B if and only if A is a subset of B without
A = B.

1 The mathematical literature of the theory of sets is very extensive. We make no
use of it beyond what will be said in the text. The interested reader will find more
information on set theory in the good introduction: A. Fraenkel : Einleitung in die Men-
genlehre, 3rd Edit. Berlin 1928; concise and technically excellent:^. Hausdorff: Mengen-
lehre, 2nd Edit. Leipzig 1927.

* If two sets A, B are both without elements, then we may say that they have the
same elements. Hence, by what we said above, A ~ B. I.e. there exists only one
empty set.

This reasoning may sound odd, but it is nevertheless faultless.

8 There are some parts of mathematics where (a) and a can be identified. This is
then occasionally done, but it is an unsound practice. It is certainly not feasible in
general. E.g., let a be something which is definitely not a one-element set, — i.e. a
two-element set («, 0), or the empty set ©. Then (a) and a must be distinguished, since
(a) is a one-element set while a is not.

62 DESCRIPTION OF GAMES OF STRATEGY

(8:A:b) The sum of two sets A, B is the set of all elements of A

together with all elements of B , — to be denoted by A u B. Simi-
larly the sums of more than two sets are formed. 1

(8:A:c) The product , or intersection , of two sets A, B is the set of all

common elements of A and of B, — to be denoted by A n B.
Similarly the products of more than two sets are formed. 1

(8:A:d) The difference of two sets A, B {A the minuend , B the subtra-

hend) is the set of all those elements of A which do not belong to
B, — to be denoted by A — B. 1

(8:A:e) When B is a subset of A, we shall also call A — B the comple-

ment of B in A. Occasionally it will be so obvious which set
A is meant that we shall simply write — B and talk about the
complement of B without any further specifications.

(8:A:f) Two sets A, B are disjunct if they have no elements in com-

mon, — i.e. if A n B = ©.

(8:A:g) A system (set) Ofc of sets is said to be a system of pairwise dis-

junct sets if all pairs of different elements of <2 are disjunct sets, —
i.e. if for A, B belonging to a, A ^ B implies A n B = ©.

8.2.3. At this point some graphical illustrations may be helpful.

We denote the objects which are elements of sets in these considerations

by dots (Figure 1). We denote sets by encircling the dots (elements)

which belong to them, writing the symbol which denotes the set across
the encircling line in one or more places (Figure 1). The sets A, C in this
figure are, by the way, disjunct, while A, B are not.

1 This nomenclature of sums, products, differences, is traditional. It is based on
certain algebraic analogies which we shall not use here. In fact, the algebra of these
operations U, n, also known as Boolean algebra , has a considerable interest of its own.
Cf. e.g. A. Tarski: Introduction to Logic, New York, 1941. Cf. further Garrett Birkhoff :
Lattice Theory, New York 1940. This book is of wider interest for the understanding of
the modern abstract method. Chapt. VI. deals with Boolean Algebras. Further litera-
ture is given there.

SETS AND PARTITIONS

63

With this device we can also represent sums, products and differences of
sets (Figure 2). In this figure neither A is a subset of B nor B one of A, —
hence neither the difference A — B nor the difference B — A is a comple-
ment. In the next figure, however, B is a subset of A, and so A — B is the
complement of B in A (Figure 3).

Figure 2. Figure 3.

8.3. Partitions, Their Properties, and Their Graphical Representation

8.3.1. Let a set ft and a system of sets ft be given. We say that &
is a partition in ft if it fulfills the two following requirements:

(8:B:a) Every element A of ft is a subset of ft, and not empty.

(8:B:b) a is a system of pairwise disjunct sets.

This concept too has been the subject of an extensive literature. 1
We say for two partitions ft, (B that ft is a subpartition of <B, if they fulfill
this condition:

(8:B:c) Every element A of ft is a subset of some element B of (B. 2

Observe that if ft is a subpartition of (B and (B a subpartition of
ft, then ft — (B. 3
Next we define:

(8 :B :d) Given two partitions ft, (B, we form the system of all those

intersections A n B — A running over all elements of ft and B over

1 Cf. G. Birkhoff loc. cit. Our requirements (8:B:a), (8:B:b) are not exactly the
customary ones. Precisely:

Ad (8:B:a): It is sometimes not required that the elements A of a be not empty.
Indeed, we shall have to make one exception in 9.1.3. (cf. footnote 4 on p. 69).

Ad (8:B:b): It is customary to require that the sum of all elements of a be exactly
the set 0. It is more convenient for our purposes to omit this condition.

2 Since a, (B are also sets, it is appropriate to compare the subset relation (as far as
a, (B are concerned) with the subpartition relation. One verifies immediately that if 0t
is a subset of (B then d is also a subpartition of (B, but that the converse statement is not
(generally) true.

3 Proof: Consider an element A of Ot. It must be subset of an element B of (B, and
B in turn subset of an element A i of Cl. So A, A\ have common elements — all those of
the not empty set A — i.e. are not disjunct. Since they both belong to the partition a,
this necessitates A *= A\. So A is a subset of B and B one of A ( = Ai). Hence A ** J5,
and thus A belongs to (B;

I.e.: a is a subset of (B. (Cf. footnote 2 above.) Similarly (B is a subset of a.
Hence a = (B.

64

DESCRIPTION OF GAMES OF STRATEGY

all those of (B — which are not empty. This again is clearly a
partition, the superposition of Ct, (B. 1

Finally, we also define the above relations for two partitions Ct, (B within
a given set C.

(8:B:e) Ct is a subpartition of (B within C, if every A belonging to Ct

which is a subset of C is also subset of some B belonging to (B
which is a subset of C.

(8:B:f) Ct is equal to (B within C if the same subsets of C are elements

of Ct and of (B.

Clearly footnote 3 on p. 63 applies again, mutatis mutandis. Also,
the above concepts within ft are the same as the original unqualified ones.

Figure 5.

8 . 3 . 2 . We give again some graphical illustrations, in the sense of 8.2.3.

We begin by picturing a partition. We shall not give the elements
of the partition — which are sets — names, but denote each one by an encir-
cling line (Figure 4).

We picture next two partitions Ct, (B distinguishing them by marking the
encircling lines of the elements of a by and of the elements of (B by

1 It is easy to show that the superposition of a, (B is a subpartition of both Ct and (B —
and that every partition e which is a subpartition of both Ct and (B is also one of their
superposition. Hence the name. Cf . G. Birkhoff , loc. cit. Chapt. I— II.

SETS AND PARTITIONS

65

— — — (Figure 5). In this figure Cl is a subpartition of (B. In the following
one neither Cl is a subpartition <B nor is <B one of Cl (Figure 6). We leave it
to the reader to determine the superposition of Cl, (B in this figure.

Figure 6.

Figure 7. Figure 8.

Figure 9.

Another, more schematic, representation of partitions obtains by repre-
senting the set G by one dot, and every element of the partition — which is a

66

DESCRIPTION OF GAMES OF STRATEGY

subset of Q — by a line going upward from this dot. Thus the partition a of
Figure 5 will be represented by a much simpler drawing (Figure 7). This
representation does not indicate the elements within the elements of the
partition, and it cannot be used to represent several partitions in ft simul-
taneously, as was done in Figure 6. However, this deficiency can be
removed if the two partitions <2, (B in ft are related as in Figure 5 : If (2 is a
subpartition of (B. In this case we can represent ft again by a dot at the
bottom, every element of (B by a line going upward from this dot — as in
Figure 7 — and every element of & as another line going further upward,
beginning at the upper end of that line of (B, which represents the element of
(B of which this element of (2 is a subset. Thus we can represent the two
partitions (2, (B of Figure 5 (Figure 8). This representation is again less
revealing than the corresponding one of Figure 5. But its simplicity makes
it possible to extend it further than pictures in the vein of Figures 4-6 could
practically go. Specifically: We can picture by this device a sequence of
partitions Cti, • • • , (2 M , where each one is a subpartition of its immediate
predecessor. We give a typical example with /z = 5 (Figure 9).

Configurations of this type have been studied in mathematics, and are
known as trees.

8.4. Logistic Interpretation of Sets and Partitions

8.4.1. The notions which we have described in 8.2.1. -8.3. 2. will be useful
in the discussion of games which follows, because of the logistic interpreta-
tion which can be put upon them.

Let us begin with the interpretation concerning sets.

If ft is a set of objects of any kind, then every conceivable property —
which some of these objects may possess, and others not — can be fully
characterized by specifying the set of those elements of ft which have this
property. I.e. if two properties correspond in this sense to the same set
(the same subset of ft), then the same elements of ft will possess these two
properties, — i.e. they are equivalent within ft, in the sense in which this term
is understood in logic.

Now the properties (of elements of ft) are not only in this simple cor-
respondence with sets (subsets of ft), but the elementary logical operations
involving properties correspond to the set operations which we discussed in
8 . 2 . 2 .

Thus the disjunction of two properties — i.e. the assertion that at least
one of them holds — corresponds obviously to forming the sum of their sets, —
the operation A u B. The conjunction of two properties — i.e. the assertion
that both hold — corresponds to forming the product of their sets, — the oper-
ation AnB. And finally, the negation of a property — i.e. the assertion
of the opposite — corresponds to forming the complement of its set, — the
operation — A. 1

1 Concerning the connection of set theory and of formal logic cf., e.g., O. Birkhoff ,
loc. cit. Chapt. VIII.

SET-THEORETICAL DESCRIPTION

67

Instead of correlating the subsets of 12 to properties in 12 — as done above
— we may equally well correlate them with all possible bodies of information
concerning an — otherwise undetermined — element of 12. Indeed, any such
information amounts to the assertion that this — unknown — element of 12
possesses a certain — specified — property. It is equivalently represented
by the set of all those elements of 12 which possess this property; i.e. to
which the given information has narrowed the range of possibilities for
the — unknown — element of 12.

Observe, in particular, that the empty set © corresponds to a property
which never occurs, i.e. to an absurd information. And two disjunct sets
correspond to two incompatible properties, i.e. to two mutually exclusive
bodies of information.

8.4.2. We now turn our attention to partitions.

By reconsidering the definition (8:B:a), (8:B:b) in 8.3.1., and by restat-
ing it in our present terminology, we see: A partition is a system of pairwise
mutually exclusive bodies of information — concerning an unknown element
of il — none of which is absurd in itself. In other words: A partition is a
preliminary announcement which states how much information will be
given later concerning an— otherwise unknown — element of 12; i.e. to what
extent the range of possibilities for this element will be narrowed later. But
the actual information is not given by the partition, — that would amount to
selecting an element of the partition, since such an element is a subset of 12,
i.e. actual information.

We can therefore say that a partition in 12 is a 'pattern of information.
As to the subsets of 12: we saw in 8.4.1. that they correspond to definite
informatibn. In order to avoid confusion with the terminology used for
partitions, we shall use in this case — i.e. for a subset of 12 — the words actual
information.

Consider now the definition (8:B:c) in 8.3.1., and relate it to our present
terminology. This expresses for two partitions d, (B in 12 the meaning of d
being a subpartition of (B : it amounts to the assertion that the information
announced by d includes all the information announced by (B (and possibly
more) ; i.e. that the pattern of information d includes the pattern of informa-
tion (B.

These remarks put the significance of the Figures 4-9 in 8.3.2. in a new
light. It appears, in particular, that the tree of Figure 9 pictures a sequence
of continually increasing patterns of information.

9. The Set-theoretical Description of a Game
9.1. The Partitions Which Describe a Game

9.1.1. We assume the number of moves — as we now know that we may —
to be fixed. Denote this number again by v, and the moves themselves
again by 9TTi, • • • , 2(TC„.

Consider all possible plays of the game T, and form the set 12 of which
they are the elements. If we use the description of the preceding sections,

68

DESCRIPTION OF GAMES OF STRATEGY

then all possible plays are simply all possible sequences <n, • • • , <r„. 1 There
exist only a finite number of such sequences, 2 * and so 12 is a finite set.

There are, however, also more direct ways to form 12. We can, e.g.,
form it by describing each play as the sequence of the v + 1 consecutive
positions 8 which arise during its course. In general, of course, a given
position may not be followed by an arbitrary position, but the positions
which are possible at a given moment are restricted by the previous posi-
tions, in a way which must be precisely described by the rules of the game. 4 *
Since our description of the rules of the game begins by forming 12, it may be
undesirable to let 12 itself depend so heavily on all the details of those rules.
We observe, therefore, that there is no objection to including in 12 absurd
sequences of positions as well. 6 * Thus it would be perfectly acceptable even
to let 12 consist of all sequences of v + 1 successive positions, without any
restrictions whatsoever.

Our subsequent descriptions will show how the really possible plays
are to be selected from this, possibly redundant, set 12.

9.1.2. v and 12 being given, we enter upon the more elaborate details of
the course of a play.

Consider a definite moment during this course, say that one which
immediately precedes a given move 9Tl«. At this moment the following
general specifications must be furnished by the rules of the game.

First it is necessary to describe to what extent the events which have
led up to the move 9TC* 6 have determined the course of the play. Every
particular sequence of these events narrows the set 12 down to a subset A K :
this being the set of all those plays from 12, the course of which is, up to 9flZ„
the particular sequence of events referred to. In the terminology of the
earlier sections, 12 is — as pointed out in 9.1.1. — the set of all sequences
a i f • • • , <t p ) then A K would be the set of those sequences <n, • • • , a, for
which the <ri, • • • , <r,_i have given numerical values (cf. footnote 6 above).
But from our present broader point of view we need only say that A K must
be a subset of 12.

Now the various possible courses the game may have taken up to 9TI,
must be represented by different sets A g . Any two such courses, if they are
different from each other, initiate two entirely disjunct sets of plays; i.e.
no play can have begun (i.e. run up to 9TC«) both ways at once. This means
that any two different sets A g must be disjunct.

1 Cf . in particular^ 6.2.2. The range of the <r lf • • • , <r„ is described in footnote 2
on p. 59.

1 Verification by means of the footnote referred to above is immediate.

* Before 3Ili, between 9Hi and 911*, between 91Z 2 and 9E«, etc., etc., between 3E r -i and
9R^, after

4 This is similar to the development of the sequence <n, • • • , <r„ as described in

footnote 2 on p. 59.

b I.e. ones which will ultimately be found to be disallowed by the fully formulated

rules of the game.

• I.e. the choices connected with the anterior moves 9Tli, • • * , 9TC«-.i — i.e. the numeri-
cal values 9 !,•••,

SET-THEORETICAL DESCRIPTION

69

Thus the complete formal possibilities of the course of all conceivable
plays of our game up to 9TC« are described by a family of pairwise disjunct
subsets of ft. This is the family of all the sets A, mentioned above. We
denote this family by Ct«.

The sum of all sets A, contained in 0t« must contain all possible plays.
But since we explicitly permitted a redundancy of ft (cf. the end of 9.1.1.),
this sum need nevertheless not be equal to ft. Summing up:

(9:A) a, is a partition in ft.

We could also say that the partition a, describes the pattern of informa-
tion of a person who knows everything that happened up to 3NI,; 1 e.g. of an
umpire who supervises the course of the play. 2

9.1.3. Second, it must be known what the nature of the move am* is
going to be. This is expressed by the k K of 6.2.1.: k K = 1, • • • , n if the
move is personal and belongs to the player k K )k K = 0 if the move is chance.
k K may depend upon the course of the play up to 3Tl„ i.e. upon the informa-
tion embodied in Cl,. 3 This means that k f must be a constant within each
set A, of &„ but that it may vary from one A, to another.

Accordingly we may form for every k = 0, 1, • • • , n a set B K (k) f which
contains all sets A, with k K = k, the various B K (k) being disjunct. Thus the
B K (k), k = 0, 1, • • • , n, form a family of disjunct subsets of ft. We denote
this family by (B,.

(9:B) (B, is again a partition in ft. Since every A, of Ct, is a subset

of some B K (k) of (B„ therefore Cl, is a subpartition of (B,.

But while there was no occasion to specify any particular enumeration
of the sets A, of ft„ it is not so with (B,. (B, consists of exactly n + 1 sets

B K (k), k = 0, 1, * * • , n, which in this way appear in a fixed enumeration
by means of the k = 0, 1, • • • , n. 4 And this enumeration is essential
since it replaces the function k K (cf. footnote 3 above).

9.1.4. Third, the conditions under which the choice connected with the
move 9m, is to take place must be described in detail.

Assume first that 9TI, is a chance move, i.e. that we are within the set
B,(0). Then the significant quantities are: the number of alternatives a,
and the probabilities p,(l), * * * , p*(ct K ) of these various alternatives (cf.
the end of 6.2.1.). As was pointed out in 7.1.1. (this was the second item

1 I.e. the outcome of all choices connected with the moves 3Ri, • • • , 9TC«-i. In our
earlier terminology: the values of <n, • • • , i.

1 It is necessary to introduce such a person since, in general, no player will be in
possession of the full information embodied in Ot*.

8 In the notations of 7.2.1., and in the sense of the preceding footnotes: k K -

fc«(<ri, • • • ,

4 Thus (B« is really not a set and not a partition, but a more elaborate concept: it con-
sists of the sets (B*(fc), fc = 0, 1, • • • , n, in this enumeration.

It possesses, however, the properties (8:B:a), (8:B:b) of 8.3.1., which characterize a
partition. Yet even there an exception must be made: among the sets (B ,(fc) there can
be empty ones.

70

DESCRIPTION OF GAMES OF STRATEGY

of the discussion there), all these quantities may depend upon the entire
information embodied in ft, (cf. footnote 3 on p. 69), since 31!, is now a
chance move. I.e. a, and the p,( 1), • • • , p K (a K ) must be constant within
each set A, of (t, 1 but they may vary from one A K to another.

Within each one of these A K the choice among the alternatives &,(1),

• • • , 6t,(a,) takes place, i.e. the choice of a <r K = 1, • • • , a K (cf. 6.2.2.).
This can be described by specifying a K disjunct subsets of A K which cor-
respond to the restriction expressed by A K , plus the choice of <r, which has
taken place. We call these sets C„ and their system — consisting of all C K
in all the A K which are subsets of 5,( 0) — 6,(0). Thus 6,(0) is a partition in

5, (0). And since every C K of 6,(0) is a subset of some A, of &„ therefore

6, (0) is a subpartition of &,.

The a K are determined by 6,(0) ; 2 hence we need not mention them any
more. For the p,( 1), • • • , p,(a,) this description suggests itself: with
every C, of 6,(0) a number p,(C,) (its probability) must be associated,
subject to the equivalents of footnote 2 on p. 50. 8

9.1.5. Assume, secondly, that 31!, is a personal move, say of the player
k = 1, * * • , n, i.e. that we are within the set B K (k). In this case we
must specify the state of information of the player k at 31!,. In 6.3.1. this
was described by means of the set A„ in 7.2.1. by means of the family of
functions <£„ the latter description being the more general and the final one.
According to this description k knows at 31!, the values of all functions
h(<n, • * * , <r,_i) of <£, and no more. This amount of information operates
a subdivision of 5,(fc) into several disjunct subsets, corresponding to the
various possible contents of k ’ s information at 3H,. We call these sets
D„ and their system £>,(&). Thus 3D, (A;) is a partition in B K (k).

Of course k’s information at 3TI, is part of the total information existing
at that moment — in the sense of 9.1.2. — which is embodied in Ct, — Hence
in an A, of (t„ which is a subset of 5,(fc), no ambiguity can exist, i.e. this
A, cannot possess common elements with more than one D, of £>,(&). This
means that the A, in question must be a subset of a D, of 3D ,(fc). In other
words: within B K (k) d K is a subpartition of 3D ,(fc).

In reality the course of the play is narrowed down at 311, within a set
A, of Ot,. But the player k whose move 31!, is, does not know as much:
as far as he is concerned, the play is merely within a set D, of 3D,(fc). He
must now make the choice among the alternatives Ct,(l), • • • , Gt,(a,), i.e.
the choice of a <r, = 1, • • • , a,. As was pointed out in 7.1.2. and 7.2.1.
(particularly at the end of 7.2.1.), a, may well be variable, but it can only
depend upon the information embodied in £>,(&). I.e. it must be a constant
within the set Z), of 3D ,(fc) to which we have restricted ourselves. Thus
the choice of a <r, = 1 , • • • , a, can be described by specifying a, disjunct
subsets of D„ which correspond to the restriction expressed by D„ plus the

1 We are within J5,( 0), hence all this refers only to A K ’s which are subsets of /?,( 0).

* a, is the number of those C, of e,(0) which are subsets of the given A,.

* I.e. every p*(C,) ^ 0, and for each A„ and the sum extended over all C, of 6,(0)
which are subsets of A, t w*e have 2p,(C,) = 1.

SET-THEORETICAL DESCRIPTION

71

choice of <r K which has taken place. We call these sets C«, and their system —
consisting of all C K in all the D K of £>«(/0 — G K (k). Thus e«(ft) is a partition
in And since every C K of C*(fc) is a subset of some of £>*(fc), there-

fore C«(fc) is a subpartition of

The a* are determined by e«(/c); 1 hence we need not mention them
any more. a K must not be zero, — i.e., given a D K of £>«(&), some C K of C«(fc),
which is a subset of must exist. 2

9.2. Discussion of These Partitions and Their Properties

9.2.1. We have completely described in the preceding sections the
situation at the moment which precedes the move 9flX«. We proceed now to
discuss what happens as we go along these moves k = 1, • • • , v. It
is convenient to add to these a k = v + 1, too, which corresponds to the
conclusion of the play, i.e. follows after the last move 9fTl„.

For k = 1, • • * , v we have, as we discussed in the preceding sections,
the partitions

a„ cb, = (£,( o), b k ( l), • • ■ , B K (n)), e«(0), e.(i), • • • , e,(n),

£>«U), * • * , £><(n).

All of these, with the sole exception of Ct*, refer to the move 3TC«, — hence
they need not and cannot be defined for k = v + 1. But (t„ + i has a per-
fectly good meaning, as its discussion in 9.1.2. shows: It represents the
full information which can conceivably exist concerning a play, — i.e. the
individual identity of the play. 3

At this point two remarks suggest themselves : In the sense of the above
observations &i corresponds to a moment at which no information is
available at all. Hence Cti should consist of the one set 12. On the other
hand, &,+! corresponds to the possibility of actually identifying the play
which has taken place. Hence 1 is a system of one-element sets.

We now proceed to describe the transition from k to k + 1, when
* = I, ‘ * * , v.

9.2.2. Nothing can be said about the change in the (B*, C«(fc), 3D*(fc)
when k is replaced by k + 1, — our previous discussions have shown that
when this replacement is made anything may happen to those objects, i.e.
to what they represent.

It is possible, however, to tell how Ct« + i obtains from Ofc«.

The information embodied in obtains from that one embodied
in Gt« by adding to it the outcome of the choice connected with the move
9Tl«. 4 This ought to be clear from the discussions of 9.1.2. Thus the

1 a K is the number of those C K of e«(fc) which are subsets of the given A K .

* We required this for k = 1, • • • , n only, although it must be equally true for
k = 0 — with an A K , subset of B K {0 ), in place of our D K of 2D*(fc). But it is unnecessary to
state it for that case, because it is a consequence of footnote 3 on p. 70; indeed, if no
C K of the desired kind existed, the 2p«(C«) of loc. cit. would be 0 and not 1.

* In the sense of footnote 1 on p. 69, the values of all <n, • • • , <r p . And the sequence
<ri, • • • , <r„ characterizes, as stated in 6.2.2., the play itself.

4 In our earlier terminology: the value of <r«.

72

DESCRIPTION OF GAMES OF STRATEGY

information in a,+i which goes beyond that in a, is precisely the information
embodied in the 6,(0), 6,(1), • • • , 6,(n).

This means that the partitions (5t,+i obtains by superposing the partition
a, with all partitions 6,(0), 6,(1), • • • , 6 ,(&)• I.e. by forming the inter-
section of every A, in Ct, with every C, in any 6,(0), 6,(1), • • • , 6,(n), and
then throwing away the empty sets.

Owing to the relationship of d K and of the 6,(fc) to the sets B,(fc) — as
discussed in the preceding sections — we can say a little more about this
process of superposition.

In B,(0), 6,(0) is a subpartition of Ct, (cf. the discussion in 9.1.4.). Hence
there &,+i simply coincides with 6,(0). In B K (k ), k = 1, • • • , n, 6 ,(fc)
and Ct, are both subpartitions of £>,(&) (cf. the discussion in 9.1.5.). Hence
there Ct,+i obtains by first taking every D, of 3D,(fc), then for every such D,
all A, of Ct, and all C, of 6 ,(fc) which are subsets of this D„ and forming all
intersections A, n C,.

Every such set A, n C, represents those plays which arise when the
player k, with the information of D, before him, but in a situation which is
really in A, (a subset of D,), makes the choice C, at the move 9TC, so as to
restrict things to C,.

Since this choice, according to what was said before, is a possible one,
there exist such plays. I.e. the set A, n C, must not be empty. We
restate this:

(9:C) If A, of Ct, and C, of 6,(fc) are subsets of the same Z), of £>,(&),

then the intersection A, n C, must not be empty.

9.2.3. There are games in which one might be tempted to set this require-
ment aside. These are games in which a player may make a legitimate
choice which turns out subsequently to be a forbidden one; e.g. the double-
blind Chess referred to in footnote 1 on p. 58: here a player can make an
apparently possible choice (“move”) on his own board, and will (possibly)
be told only afterwards by the “umpire” that it is an “impossible” one.

This example is, however, spurious. The move in question is best
resolved into a sequence of several alternative ones. It seems best to give
the contemplated rules of double-blind Chess in full.

The game consists of a sequence of moves. At each move the “umpire ”
announces to both players whether the preceding move was a “possible”
one. If it was not, the next move is a personal move of the same player
as the preceding one; if it was, then the next move is the other player’s
personal move. At each move the player is informed about all of his own
anterior choices, about the entire sequence of “possibility” or “impossibil-
ity” of all anterior choices of both players, and about all anterior instances
where either player threatened check or took anything. But he knows
the identity of his own losses only. In determining the course of the game,
the “umpire” disregards the “impossible” moves. Otherwise the game is
played like Chess, with a stop rule in the sense of footnote 3 on p. 59,

AXIOMATIC FORMULATION

73

amplified by the further requirement that no player may make (“try”)
the same choice twice in any one uninterrupted sequence of his own personal
moves. (In practice, of course, the players need two chessboards — out of
each other’s view but both in the “umpire’s” view — to obtain these condi-
tions of information.)

At any rate we shall adhere to the requirement stated above. It will
be seen that it is very convenient for our subsequent discussion (cf. 11.2.1.).

9.2.4. Only one thing remains: to reintroduce in our new terminology,
the quantities SF*, k = 1, • • • , n, of 6.2.2. $F* is the outcome of the play

for the player k . $F* must be a function of the actual play which has taken

place. 1 If we use the symbol t to indicate that play, then we may say:
SF k is a function of a variable tt with the domain of variability 12. I.e. :

£Ffc = $F/c(7r), it m 12, k = 1, * * * ,

10. Axiomatic Formulation
10.1. The Axioms and Their Interpretations

10.1.1. Our description of the general concept of a game, with the new
technique involving the use of sets and of partitions, is now complete.
All constructions and definitions have been sufficiently explained in the
past sections, and we can therefore proceed to a rigorous axiomatic definition
of a game. This is, of course, only a concise restatement of the things
which we discussed more broadly in the preceding sections.

We give first the precise definition, without any commentary: 2
An n- person game T, i.e. the complete system of its rules, is determined
by the specification of the following data:

(10:A:a) A number v.

(10:A:b) A finite set 12.

(10:A:c) For every k = 1, • * • , n: A function

— — Jit (tt) , it in 12.

(10:A:d) For every k = 1, • • • , v, v + 1: A partition in 12.

(10:A:e) For every k = 1, • • • , v: A partition (B« in 12. ©« con-
sists of n + 1 sets B K (k), k = 0, 1, • • • , n, enumerated in

this way.

(10:A:f) For every k = 1, * • • , v and every k = 0, 1, • • • , n:

A partition (3, (k) in B K (k).

(10:A:g) For every k = 1, • • • , v and every k = 1, • • • , n: A

partition £>*(fc) in B K (k).

(10:A:h) For every k = 1, • • • , v and every C K of C*(0): A number

P*(C<).

These entities must satisfy the following requirements:

(10:1 :a) is a subpartition of ©«.

(10:1 :b) C«(0) is a subpartition of 0t«.

1 In the old terminology, accordingly, we had g* «* g*(a i, • • • , o>). Cf. 6.2.2.

’For “explanations” cf. the end of 10.1.1. and the discussion of 10.1.2.

74

DESCRIPTION OF GAMES OF STRATEGY

(10:1 :c) For k = 1, • • • , n: <5 K (k) is a subpartition of $>«(fc).

(10:1 :d) For k = 1, • • • , n: Within B*(fc), 0t< is a subpartition of

£>,(*).

(10:1 :e) For every k = 1, • • • , v and every A K of 0t< which is a

subset of B k ( 0): For all C K of 6,(0) which are subsets of this
A K , Pk(C k ) ^ 0, and for the sum extended over them 2p*(C,) = 1.
(10:1 :f) &i consists of the one set Q.

(10:l:g) G„+i consists of one-element sets.

(10:1 :h) For k = 1, • • • , v: <2,+i obtains from Ct K by superposing it

with all 6*(fc), k = 0, 1, • • • , n. (For details, cf. 9.2.2.)

(10:1 :i) For k = 1, • • • , v: If A K of d K and C K of 6 *(fc), k = 1,

• • • , n are subsets of the same D K of £>*(&)> then the inter-

section A k n C K must not be empty.

(10:1 :j) For k = 1, • • • , v and k = 1, • • • , n and every D K of

3D K (k ) : Some C K (k) of e K , which is a subset of D K , must exist.

This definition should be viewed primarily in the spirit of the modern
axiomatic method. We have even avoided giving names to the mathe-
matical concepts introduced in (10:A:a)-(10:A:h) above, in order to estab-
lish no correlation with any meaning which the verbal associations of names
may suggest. In this absolute “ purity ” these concepts can then be the
objects of an exact mathematical investigation. 1

This procedure is best suited to develop sharply defined concepts.
The application to intuitively given subjects follows afterwards, when
the exact analysis has been completed. Cf. also what was said in 4.1.3.
in Chapter I about the role of models in physics: The axiomatic models
for intuitive systems are analogous to the mathematical models for (equally
intuitive) physical systems.

Once this is understood, however, there can be no harm in recalling
that this axiomatic definition was distilled out of the detailed empirical
discussions of the sections, which precede it. And it will facilitate its use,
and make its structure more easily understood, if we give the intervening
concepts appropriate names, — which indicate, as much as possible, the
intuitive background. And it is further useful to express, in the same
spirit, the “meaning” of our postulates (10:1 :a)-(10:l :j) — i.e. the intuitive
considerations from which they sprang.

All this will be, of course, merely a concise summary of the intuitive con-
siderations of the preceding sections, which lead up to this axiomatization.

10 . 1 . 2 . We state first the technical names for the concepts of (10:A:a)-
(10 :A :h) in 10.1.1.

1 This is analogous to the present attitude in axiomatizing such subjects as logic,
geometry, etc. Thus, when axiomatizing geometry, it is customary to state that the
notions of points, lines, and planes are not to be a priori identified with anything intui-
tive, — they are only notations for things about which only the properties expressed in
the axioms are assumed. Cf., e.g., D. Hilbert : Die Grundlagen der Geometrie, Leipzig
1899, 2nd Engl. Edition Chicago 1910.

AXIOMATIC FORMULATION

75

(10:A:a*)

(10:A:b*)

(10:A:c*)

(10:A:d*)

(10:A:e*)

(10:A:f*)

(10 :A :g*)
(10 :A :h*)

v is the length of the game T.

ft is the set of all plays of T.

(Ffc^r) is the outcome of the play w for the player k.

Ot* is the umpire’s pattern of information , an A * of Ot* is the
umpire’s actual information at (i.e. immediately preceding) the
move 9TI*. (For k = v + 1 : At the end of the game.)

(B* is the pattern of assignment ) a B K (k) of (B* is the actual
assignment, of the move 9TI*.

6*(fc) is the pattern of choice, a C* of 6*(fc) is the actual
choice, of the player k at the move 911*. (For k = 0: Of
chance.)

£)*(&) is the player k’s pattern of information, a D K of £>*(fc)
the player k’s actual information, at the move 9TI*.

Pk(C k ) is the probability of the actual choice C* at the
(chance) move 3TI*.

We now formulate the “meaning” of the requirements (10:1 :a)-
(10:1 :j) — in the sense of the concluding discussion of 10.1.1 — with the use of
the above nomenclature.

(10:1 :a*)
(10:1 :b*)
(10:1 :c*)
(10:1 :d*)

(10:1 :e*)

(10:1 :f*)
(10:l:g*)
(10:1 :h*)

(10:1 :i*)

The umpire’s pattern of information at the move 9TC*
includes the assignment of that move.

The pattern of choice at a chance move 9TI* includes the
umpire’s pattern of information at that move.

The pattern of choice at a personal move 3TI* of the player k
includes the player k’s pattern of information at that move.

The umpire’s pattern of information at the move 9TI*
includes — to the extent to which this is a personal move of the
player k — the player k’s pattern of information at that move.

The probabilities of the various alternative choices at a
chance move 9TI* behave like probabilities belonging to disjunct
but exhaustive alternatives.

The umpire’s pattern of information at the first move is
void.

The umpire’s pattern of information at the end of the game
determines the play fully.

The umpire’s pattern of information at the move 9Tl* + i
(for k = v: at the end of the game) obtains from that one at
the move 9TI* by superposing it with the pattern of choice at
the move 9TI*.

Let a move 9Tt* be given, which is a personal move of the
player k, and any actual information of the player k at that
move also be given. Then any actual information of the
umpire at that move and any actual choice of the player k at
that move, which are both within (i.e. refinements of) this
actual (player’s) information, are also compatible with each

Othe r T ** fcViPv nppnr in npt.iifll nlnvs

76

DESCRIPTION OF GAMES OF STRATEGY

(10:1 :j*) Let a move 911* be given, which is a personal move of the

player k , and any actual information of the player k at that
move also be given. Then the number of alternative actual
choices, available to the player k, is not zero.

This concludes our formalization of the general scheme of a game.

10.2. Logistic Discussion of the Axioms

10 . 2 . We have not yet discussed those questions which are convention-
ally associated in formal logics with every axiomatization: freedom from
contradiction, categoricity (completeness), and independence of the axioms. 1
Our system possesses the first and the last-mentioned properties, but not the
second one. These facts are easy to verify, and it is not difficult to see
that the situation is exactly what it should be. In summa:

Freedom from contradiction: There can be no doubt as to the existence
of games, and we did nothing but give an exact formalism for them. We
shall discuss the formalization of several games later in detail, cf. e.g. the
examples of 18., 19. From the strictly mathematical — logistic — point of
view, even the simplest game can be used to establish the fact of freedom
from contradiction. But our real interest lies, of course, with the more
involved games, which are the really interesting ones. 2

Categoricity (completeness): This is not the case, since there exist
many different games which fulfill these axioms. Concerning effective
examples, cf. the preceding reference.

The reader will observe that categoricity is not intended in this case,
since our axioms have to define a class of entities (games) and not a unique
entity. 8

Independence: This is easy to establish, but we do not enter upon it.

10.3. General Remarks Concerning the Axioms

10 . 3 . There are two more remarks which ought to be made in connection
with this axiomatization.

First, our procedure follows the classical lines of obtaining an exact
formulation for intuitively — empirically — given ideas. The notion of a
game exists in general experience in a practically satisfactory form, which is
nevertheless too loose to be fit for exact treatment. The reader who has
followed our analysis will have observed how this imprecision was gradually

1 Cf. D. Hilbert, loc. cit. ; 0. Veblen & J. W. Young: Projective Geometry, New York
1910; H. Weyl: Philosophic der Mathematik und Naturwissenschaften, in Handbuch der
Philosophic, Munich, 1927.

1 This is the simplest game: v - 0, 12 has only one element, say7To. Consequently
no (B«, <S«(fc), Dic(fc), exist, while the only 0* is di, consisting of 12 alone. Define ff(iro) =» 0
for A ■* 1, • • • , ». An obvious description of this game consists in the statement that
nobody does anything and that nothing happens. This also indicates that the freedom
from contradiction is not in this case an interesting question.

8 This is an important distinction in the general logistic approach to axiomatization.
Thus the axioms of Euclidean geometry describe a unique object — while those of group
theory (in mathematics) or of rational mechanics (in physics) do not, since there exist
many different groups and many different mechanical systems.

AXIOMATIC FORMULATION 77

removed, the “zone of twilight” successively reduced, and a precise formula-
tion obtained eventually.

Second, it is hoped that this may serve as an example of the truth of a
much disputed proposition: That it is possible to describe and discuss
mathematically human actions in which the main emphasis lies on the
psychological side. In the present case the psychological element was
brought in by the necessity of analyzing decisions, the information on the
basis of which they are taken, and the interrelatedness of such sets of
information (at the various moves) with each other. This interrelatedness
originates in the connection of the various sets of information in time,
causation, and by the speculative hypotheses of the players concerning
each other.

There are of course many — and most important — aspects of psychology
which we have never touched upon, but the fact remains that a primarily
psychological group of phenomena has been axiomatized.

10.4. Graphical Representation

10 . 4 . 1 . The graphical representation of the numerous partitions which
we had to use to represent a game is not easy. We shall not attempt to
treat this matter systematically: even relatively simple games seem to
lead to complicated and confusing diagrams, and so the usual advantages of
graphical representation do not obtain.

There are, however, some restricted possibilities of graphical representa-
tion, and we shall say a few words about these.

In the first place it is clear from (10:1 :h) in 10.1.1., (or equally by
(10:1 :h*) in 10.1.2., i.e. by remembering the “meaning,”) that is a
subpartition of Ct K . I.e. in the sequence of partitions Oti, • • • , Ct,, Ct,+i
each one is a subpartition of its immediate predecessor. Consequently this
much can be pictured with the devices of Figure 9 in 8.3.2., i.e. by a tree.
(Figure 9 is not characteristic in one way: since the length of the game T is
assumed to be fixed, all branches of the tree must continue to its full height.
Cf. Figure 10 in 10.4.2. below.) We shall not attempt to add the B K (k) f
& K (k), £)«(&) to this picture.

There is, however, a class of games where the sequence Oti, • • • , 0t,,0t,+i
tells practically the entire story. This is the important class — already
discussed in 6.4.1., and about which more will be said in 15. — where
preliminarily and anteriority are equivalent. Its characteristics find a
simple expression in our present formalism.

10 . 4 . 2 . Preliminarity and anteriority are equivalent — as the discussions
of 6.4.1., 6.4.2. and the interpretation of 6.4.3. show — if and only if every
player who makes a personal move knows at that moment the entire anterior
history of the play. Let the player be fc, the move 9Tl«. The assertion
that 9fTl, is k ’ s personal move means, then, that we are within B K (k). Hence
the assertion is that within B K {k ) the player k’s pattern of information
coincides with the umpire’s pattern of information; i.e. that 2D«(fc) is equal to

78

DESCRIPTION OF GAMES OF STRATEGY

d K within B K (k). But £>«(&) is a partition in B K (k); hence the above state-
ment means that £>,(&) simply is that part of C i K which lies in B K (k).

We restate this:

(10:B) Preliminarity and anteriority coincide — i.e. every player who

makes a personal move is at that moment fully informed about
the entire anterior history of the play — if and only if £> K (k) is
that part of C l K which lies in B K (k).

If this is the case, then we can argue on as follows: By (10:1 :c) in 10.1.1.
and the above, Q K (k) must now be a subpartition of d K . This holds for
personal moves, i.e. for A; = 1, • • • , n, but for k = 0 it follows immedi-

ately from (10:1 :b) in 10.1.1. Now (10:1 :h) in 10.1.1. permits the inference
from this (for details cf. 9.2.2.) that Ct*+i coincides with (B* (k) in B K (k) — for
all k = 0, 1 , • • • , n. (We could equally have used the corresponding
points in 10.1.2., i.e. the “meaning” of these concepts. We leave the verbal
expression of the argument to the reader.) But Q K (k) is a partition in B K (k ) ;
hence the above statement means that e« (k) simply is that part of Gfc«+i
which lies in B K (k).

We restate this:

(10 :C) If the condition of (10 :B) is fulfilled, then Q K (k) is that part

of G*+] which lies in B K (k).

Thus when preliminarity and anteriority coincide, then in our present
formalism the sequence (&!,-••, &y,&v+i and the sets B K (k), k = 0, 1,
• * * , n, for each k = 1 , • * • , v> describe the game fully. I.e. the picture

STRATEGIES AND FINAL SIMPLIFICATION

79

of Figure 9 in 8.3.2. must be amplified only by bracketing together those
elements of each (5t«, which belong to the same set (B *(fc). (Cf. however, the
remark made in 10.4.1.) We can do this by encircling them with a line,
across which the number k of B K (k) is written. Such B K (k) as are empty can
be omitted. We give an example of this for v = 5 and n = 3 (Figure 10).

In many games of this class even this extra device is not necessary,
because for every k only one B K (k) is not empty. I.e. the character of each
move 9Tl« is independent of the previous course of the play. 1 Then it
suffices to indicate at each d K the character of the move 9TC* — i.e, the unique
k = 0, 1, • • • , n for which B K (k) ©.

11. Strategies and the Final Simplification of the Description of a Game
11.1. The Concept of a Strategy and Its Formalization

11.1.1. Let us return to the course of an actual play t of the game T.

The moves 9 TC* follow each other in the order k = 1 , • • • , v. At each
move 9Tl« a choice is made, either by chance — if the play is in B K ( 0) — or by a
player k = 1, • • • , n — if the play is in B K (k). The choice consists in the
selection of a C K from e«(fc) (k — 0 or k = 1 , • • • , n, cf. above), to which
the play is then restricted. If the choice is made by a player k, then pre-
cautions must be taken that this player’s pattern of information should be
at this moment £>*(&), as required. (That this can be a matter of some
practical difficulty is shown by such examples as Bridge [cf. the end of
6.4.2.] and double-blind Chess [cf. 9.2.3.].)

Imagine now that each player k = 1, • • • , n, instead of making each
decision as the necessity for it arises, makes up his mind in advance for all
possible contingencies; i.e. that the player k begins to play with a complete
plan: a plan which specifies what choices he will make in every possible situa-
tion, for every possible actual information which he may possess at that
moment in conformity with the pattern of information which the rules of
the game provide for him for that case. We call such a plan a strategy.

Observe that if we require each player to start the game with a complete
plan of this kind, i.e. with a strategy, we by no means restrict his freedom
of action. In particular, we do not thereby force him to make decisions
on the basis of less information than there would be available for him in each
practical instance in an actual play. This is because the strategy is sup-
posed to specify every particular decision only as a function of just that
amount of actual information which would be available for this purpose in
an actual play. The only extra burden our assumption puts on the player
is the intellectual one to be prepared with a rule of behavior for all even-
tualities, — although he is to go through one play only. But this is an innoc-
uous assumption within the confines of a mathematical analysis. (Cf.
also 4.1.2.)

1 This is true for Chess. The rules of Backgammon permit interpretations both
w^ys.

80 DESCRIPTION OF GAMES OF STRATEGY

11 . 1 . 2 . The chance component of the game can be treated in the same
way.

It is indeed obvious that it is not necessary to make the choices which are
left to chance, i.e. those of the chance moves, only when those moves come
along. An umpire could make them all in advance, and disclose their
outcome to the players at the various moments and to the varying extent,
as the rules of the game provide about their information.

It is true that the umpire cannot know in advance which moves will be
chance ones, and with what probabilities; this will in general depend upon
the actual course of the play. But — as in the strategies which we considered
above — he could provide for all contingencies: He could decide in advance
what the outcome of the choice in every possible chance move should be, for
every possible anterior course of the play, — i.e. for every possible actual
umpire’s information at the move in question. Under these conditions the
probabilities prescribed by the rules of the game for each one of the above
instances would be fully determined — and so the umpire could arrange for
each one of the necessary choices to be effected by chance, with the appro-
priate probabilities.

The outcomes could then be disclosed by the umpire to the players — at
the proper moments and to the proper extent — as described above.

We call such a preliminary decision of the choices of all conceivable
chance moves an umpire’s choice.

We saw in the last section that the replacement of the choices of all
personal moves of the player k by the strategy of the player k is legitimate;
i.e. that it does not modify the fundamental character of the game T.
Clearly our present replacement of the choices of all chance moves by the
umpire’s choice is legitimate in the same sense.

11 . 1 . 3 . It remains for us to formalize the concepts of a strategy and of
the umpire’s choice. The qualitative discussion of the two last sections
makes this an unambiguous task.

A strategy of the player k does this: Consider a move 9fld„. Assume that
it has turned out to be a personal move of the player — i.e. assume that
the play is within B K (k). Consider a possible actual information of the
player k at that moment, — i.e. consider a D« of 3D*(fc). Then the strategy
in question must determine his choice at this juncture, — i.e. a C K of ©,(&)
which is a subset of the above D K .

Formalized:

(11 :A) A strategy of the player k is a function 2 *(k; D k ) which is

defined for every k = 1, • • • , v and every D K of £>« (k), and
whose value

2*(k; D k ) = C K

has always these properties: C K belongs to e«(fc) and is a subset
of D k .

That strategies — i.e. functions 2 *(k; D k ) fulfilling the above requirement
— exist at all, coincides precisely with our postulate (10:1 :j) in 10.1.1.

STRATEGIES AND FINAL SIMPLIFICATION

81

An umpired choice does this:

Consider a move 9TI,. Assume that it has turned out to be a chance
move, — i.e. assume that the play is within B,( 0). Consider a possible
actual information of the umpire at this moment; i.e. consider an A, of Cl,
which is a subset of B,(0). Then the umpire’s choice in question must
determine the chance choice at this juncture, — i.e. a C, of 6,(0) which is a
subset of the above A K .

Formalized :

(11 :B) An umpires choice is a function 2 0 (#c; A K ) which is defined for

every k = 1, • • • , v and every A, of Ct, which is a subset of
B,(0) and whose value

2o(*; A,) = C,

has always these properties: C K belongs to 6,(0) and is a subset
of A,.

Concerning the existence of umpire’s choices — i.e. of functions 2 0 (*; A,)
fulfilling the above requirement — cf. the remark after (11 :A) above, and
footnote 2 on p. 71.

Since the outcome of the umpire’s choice depends on chance, the cor-
responding probabilities must be specified. Now the umpire’s choice is an
aggregate of independent chance events. There is such an event, as
described in 11.1.2., for every k = 1, • • • , v and every A, of ft, which is a
subset of J3,(0). I.e. for every pair k, A, in the domain of definition of
2 0 (k; A,). As far as this event is concerned the probability of the particular
outcome 2 0 (*; A,) = C, is p,(C«). Hence the probability of the entire
umpire’s choice, represented by the function 2 0 (/c; A,) is the product of the
individual probabilities p K (C K ). 1

Formalized:

(11 :C) The probability of the umpire’s choice , represented by the

function 2 0 (k; A,) is the product of the probabilities p,(C,),
where 2 0 (k; A,) = C„ and k, A, run over the entire domain of
definition of 2 0 (k; A,) (cf. (11 :B) above).

If we consider the conditions of (10:1 :e) in 10.1.1. for all these pairs
k, A„ and multiply them all with each other, then these facts result: The
probabilities of (11 :C) above are all ^ 0, and their sum (extended over all
umpire’s choices) is one. This is as it should be, since the totality of all
umpire’s choices is a system of disjunct but exhaustive alternatives.

ll.). The Final Simplification of the Description of a Game

11.2.1. If a definite strategy has been adopted by each player k = 1,
• • • , n, and if a definite umpire’s choice has been selected, then these
determine the entire course of the play uniquely, — and accordingly its

1 The chance events in question must be treated as independent.

82

DESCRIPTION OF GAMES OF STRATEGY

outcome too, for each player k = 1, • • • , n. This should be clear from
the verbal description of all these concepts, but an equally simple formal
proof can be given.

Denote the strategies in question by 2 *(k; D K ), k = 1, • • • , n, and the
umpired choice by 2 0 (k; A,). We shall determine the umpired actual
information at all moments k = 1, • • • , v } v + 1. In order to avoid
confusing it with the above variable A„ we denote it by A*.

Ai is, of course, equal to ft itself. (Cf. (10:1 :f) in 10.1.1.)

Consider now a k = 1, • • • , v ) and assume that the corresponding A*
is already known. Then A« is a subset of precisely one B K (k), k =0, 1,

• • • , n. (Cf. (10:1 :a) in 10.1.1.) If k = 0, then 3TI* is a chance move, and
so the outcome of the choice is 2 0 (k; A*). Accordingly A*+i = 2 0 (k; A,).
(Cf. (10:1 :h) in 10.1.1. and the details in 9.2.2.) If k = 1, • • • , n, then
3TC, is a personal move of the player k. A* is a subset of precisely one D K of
3D«(fc). (Cf. (10:1 :d) in 10.1.1.) So the outcome of the choice is 2 *(k; D k ).
Accordingly A,+i = A* n 2*(*; D K ). (Cf. (10:1 :h) in 10.1.1. and the details
in 9.2.2.)

Thus we determine inductively A h A 2 , A 3 , * * * , A„ A v + 1 in succession.
But A „+ 1 is a one-element set (cf. (10:1 :g) in 10.1.1.); denote its unique
element by ft.

This ft is the actual play which took place. 1 Consequently the outcome
of the play is $*(#) for the player k = 1,* * * , n.

11.2.2. The fact that the strategies of all players and the umpire's
choice determine together the actual play — and so its outcome for each
player — opens up the possibility of a new and much simpler description of
the game T.

Consider a given player k = 1, • • • , n. Form all possible strategies
of his, 2 *(k; Z),), or for short 2*. While their number is enormous,
it is obviously finite. Denote it by 0*, and the strategies themselves by

Form similarly all possible umpire's choices, 2 0 (k; A«), or for short 2 0 .
Again their number is finite. Denote it by /3 0 , and the umpire's choices by
2j, * • • , 2?o. Denote their probabilities by p 1 , • • • , respectively.
(Cf. (11 :C) in 11.1.3.) All these probabilities are ^ 0 and their sum is one.
(Cf. the end of 11.1.3.)

A definite choice of all strategies and of the umpire's choices, say 2J* for
k = 1, • • • , n and for k = 0 respectively, where

r* = 1, • • • , 0* for k = 0, 1, • • • , n,

determines the play ft (cf. the end of 11.2.1.), and its outcome $h(ft) for
each player k = 1, * • • , n. Write accordingly

(11:1) $k(ft) = 9*( r o> ti, • • • , r„) for k = 1, • • • , n.

1 The above inductive derivation of the Ai, A 2 , A s, • • • , A, f A v +i is just a mathemat-
ical reproduction of the actual course of the play. The reader should verify the parallel-
ism of the steps involved.

STRATEGIES AND FINAL SIMPLIFICATION

83

The entire play now consists of each player k choosing a strategy 2J*,
i.e. a number i * = 1, • • • , 0*; and of the chance umpire’s choice of r 0 = 1,

• • • , 0o, with the probabilities p 1 , • • • , p*o respectively.

The player k must choose his strategy, i.e. his r*, without any information
concerning the choices of the other players, or of the chance events (the
umpire’s choice). This must be so since all the information he can at any
time possess is already embodied in his strategy 2 fc = 2J* i.e. in the function
2* = 2 *(k; D k ). (Cf. the discussion of 11.1.1.) Even if he holds definite
views as to what the strategies of the other players are likely to be, they
must be already contained in the function 2* (k; D«).

11.2.3. All this means, however, that V has been brought back to the
very simplest description, within the least complicated original framework of
the sections 6.2. 1.-6. 3.1. We have n + 1 moves, one chance and one
personal for each player k = 1 , • • • , n — each move has a fixed number of
alternatives, 0o for the chance move and 0i, • • • , 0„ for the personal ones —
and every player has to make this choice with absolutely no information
concerning the outcome of all other choices. 1

Now we can get rid even of the chance move. If the choices of the
players have taken place, the player k having chosen r*, then the total
influence of the chance move is this : The outcome of the play for the player k
may be any one of the numbers

0) 7 0 = lj * * * > 00)

with the probabilities p 1 , • • * , p^o respectively. Consequently his
“mathematical expectation” of the outcome is

B 0

(11:2) 3C*(ti, • • • ,t„) = P r ° S*( T o, ti, • • • , r»).

r 0 -l

The player’s judgment must be directed solely by this “mathematical
expectation,” — because the various moves, and in particular the chance
move, are completely isolated from each other. 2 Thus the only moves
which matter are the n personal moves of the players k = 1, • • • n.

The final formulation is therefore this:

(11 :D) The n person game T, i.e. the complete system of its rules, is

determined by the specification of the following data:

(ll:D:a) For every k = 1, • • • , n: A number 0*.

(Il:D:b) For every k = 1, • • • , n: A function

3C * = 3C*(ri, • • , r n ),

r, = 1, • * * , 0/ for j = 1, • • • , n.

1 Owing to this complete disconnectedness of the n + 1 moves, it does not matter
in what chronological order they are placed.

2 We are entitled to use the unmodified “mathematical expectation” since we are
satisfied with the simplified concept of utility, as stressed at the end of 5.2.2 This
excludes in particular all those more elaborate concepts of “expectation,” which are
really attempts at improving that naive concept of utility. (E.g. D. Bernoulli's “moral
expectation” in the “St. Petersburg Paradox.”)

84

DESCRIPTION OF GAMES OF STRATEGY

The course of a play of T is this:

Each player k chooses a number t* = 1, • • • , /J*. Each
player must make his choice in absolute ignorance of the choices
of the others. After all choices have been made, they are
submitted to an umpire who determines that the outcome of the
play for the player k is 3C*(ri, • • • , r„).

11.3. The Role of Strategies in the Simplified Form of a Game

11 . 3 . Observe that in this scheme no space is left for any kind of further
“strategy." Each player has one move, and one move only; and he must
make it in absolute ignorance of everything else. 1 This complete crystal-
lization of the problem in this rigid and final form was achieved by our
manipulations of the sections from 11.1.1. on, in which the transition from
the original moves to strategies was effected. Since we now treat these
strategies themselves as moves, there is no need for strategies of a higher
order.

11.4. The Meaning of the Zero-sum Restriction

11 . 4 . We conclude these considerations by determining the place of the
zero-sum games (cf. 5.2.1.) within our final scheme.

That T is a zero-sum game means, in the notation of 10.1.1., this:

n

(11:3) = 0 for a N T °f R

t-i

If we pass from ?*(*•) to 8*( T o, n, • • • , r n ), in the sense of 11.2.2., then this
becomes

n

(11:4) 9*( t o> t i> • • • , r n ) = 0 forallro.n, • • • , t„.

*-i

And if we finally introduce 3C*(n, • • • , t„), in the sense of 11.2.3., we obtain

n

(11:5) 2) , t„) = 0 for all n, • • • , r n .

*-i

Conversely, it is clear that the condition (11:5) makes the game T, which we
defined in 11.2.3., one of zero sum.

1 Reverting to the definition of a strategy as given in 11.1.1.: In this game a player &
has one and only one personal move, and this independently of the course of the play,
— the move 911*. And he must make his choice at 911* with nil information. So his
strategy is simply a definite choice for the move 9TI*, — no more and no less; i.e. precisely
n - 1 , • • , 0 *.

We leave it to the reader to describe this game in terms of partitions, and to compare
the above with the formalistic definition of a strategy in (11 :A) in 11.1.3.

CHAPTER III

ZERO-SUM TWO-PERSON GAMES: THEORY

12. Preliminary Survey
12.1. General Viewpoints

12.1.1. In the preceding chapter we obtained an all-inclusive formal
characterization of the general game of n persons (cf. 10.1.). We followed
up by developing an exact concept of strategy which permitted us to replace
the rather complicated general scheme of a game by a much more simple
special one, which was nevertheless shown to be fully equivalent to the
former (cf. 11.2.). In the discussion which follows it will sometimes be
more convenient to use one form, sometimes the other. It is therefore
desirable to give them specific technical names. We will accordingly call
them the extensive and the normalized form of the game, respectively.

Since these two forms are strictly equivalent, it is entirely within our
province to use in each particular case whichever is technically more con-
venient at that moment. We propose, indeed, to make full use of this
possibility, and must therefore re-emphasize that this does not in the least
affect the absolutely general validity of all our considerations.

Actually the normalized form is better suited for the derivation of
general theorems, while the extensive form is preferable for the analysis of
special cases; i.e., the former can be used advantageously to establish pro-
perties which are common to all games, while the latter brings out charac-
teristic differences of games and the decisive structural features which
determine these differences. (Cf. for the former 14., 17., and for the latter
e.g. 15.)

12.1.2. Since the formal description of all games has been completed,
we must now turn to build up a positive theory. It is to be expected that
a systematic procedure to this end will have to proceed from simpler games
to more complicated games. It is therefore desirable to establish an
ordering for all games according to their increasing degree of complication.

We have already classified games according to the number of partici-
pants — a game with n participants being called an n- person game — and
also according to whether they are or are not of zero-sum. Thus we must
distinguish zero-sum n-person games and general n-person games. It will
be seen later that the general n- person game is very closely related to the
zero-sum (n +» l)-person game, — in fact the theory of the former will
obtain as a special case of the theory of the latter. (Cf. 56.2.2.)

12.2. The One-person Game

12.2.1. We begin with some remarks concerning the one-person game.
In the normalized form this game consists of the choice of a number

85

86

ZERO-SUM TWO-PERSON GAMES: THEORY

r = 1, • • • , /?, after which the (only) player 1 gets the amount 3C(r). 1 The
zero-sum case is obviously void 2 and there is nothing to say concerning it.
The general case corresponds to a general function 3C(r) and the “best”
or “rational” way of acting — i.e. of playing — consists obviously of this:
The player 1 will choose r = 1, • • • , $ so as to make 5C(r) a maximum.

This extreme simplification of the one-person game is, of course, due
to the fact that our variable r represents not a choice (in a move) but
the player’s strategy; i.e., it expresses his entire “theory” concerning the
handling of all conceivable situations which may occur in the course of the
play. It should be remembered that even a one-person game can be of a
very complicated pattern: It may contain chance moves as well as personal
moves (of the only player), each one possibly with numerous alternatives,
and the amount of information available to the player at any particular
personal move may vary in any prescribed way.

12 . 2 . 2 . Numerous good examples of many complications and subtleties
which may arise in this way are given by the various games of “Patience”
or “Solitaire.” There is, however, an important possibility for which, to
the best of our knowledge, examples are lacking among the customary
one-person games. This is the case of incomplete information, i.e. of non-
equivalence of anteriority and preliminarity of personal moves of the
unique player (cf. 6.4.). For such an absence of equivalence it would be
necessary that the player have two personal moves SfTC* and 9Tt x at neither
of which he is informed about the outcome of the choice of the other.
Such a state of lack of information is not easy to achieve, but we discussed
in 6.4.2. how it can be brought about by “splitting” the player into two
or more persons of identical interest and with imperfect communications.
We saw loc. cit. that Bridge is an example of this in a two-person game;
it would be easy to construct an analogous one-person game — but unluckily
the known forms of “solitaire” are not such. 8

This possibility is nevertheless a practical one for certain economic
setups: A rigidly established communistic society, in which the structure
of the distribution scheme is beyond dispute (i.e. where there is no exchange,
but only one unalterable imputation) would be such — since the interests
of all the members of such a society are strictly identical 4 this setup must be
treated as a one-person game. But owing to the conceivable imperfections
of communications among the members, all sorts of incomplete information
can occur.

This is then the case which, by a consistent use of the concept of strategy
(i.e. of planning), is naturally reduced to a simple maximum problem. On
the basis of our previous discussions it will therefore be apparent now

1 Cf. (ll:D:a), (ll:D:b) at the end of 11.2.3. We suppress the index 1.

* Then 3C(r) = 0, cf. 11.4.

3 The existing “ double solitaires” are competitive games between the two partici-
pants, i.e. two-person games.

4 The individual members themselves cannot be considered as players, since all
possibilities of conflict among them, as well as coalitions of some of them against the
others, are excluded.

PRELIMINARY SURVEY 87

that this — and this only — is the case in which the simple maximum formu-
lation — i.e. the “ Robinson Crusoe” form — of economics is appropriate.

12 . 2 . 3 . These considerations show also the limitations of the pure
maximum — i.e. the “Robinson Crusoe” — approach. The above example
of a society of a rigidly established and unquestioned distribution scheme
shows that on this plane a rational and critical appraisal of the distribution
scheme itself is impossible. In order to get a maximum problem it was
necessary to place the entire scheme of distribution among the rules of the
game, which are absolute, inviolable and above criticism. In order to
bring them into the sphere of combat and competition — i.e. the strategy
of the game — it is necessary to consider n- person games with n ^ 2 and
thereby to sacrifice the simple maximum aspect of the problem.

12.3. Chance and Probability

12 . 3 . Before going further, we wish to mention that the extensive
literature of “mathematical games” — which was developed mainly in
the 18th and 19th centuries — deals essentially only with an aspect of the
matter which we have already left behind. This is the appraisal of the
influence of chance. This was, of course, effected by the discovery and
appropriate application of the calculus of probability and particularly
of the concept of mathematical expectations. In our discussions, the
operations necessary for this purpose were performed in 11.2.3. 1,2

Consequently we are no longer interested in these games, where the
mathematical problem consists only in evaluating the role of chance — i.e.
in computing probabilities and mathematical expectations. Such games
lead occasionally to interesting exercises in probability theory; 8 but we
hope that the reader will agree with us that they do not belong in the
theory of games proper.

12.4. The Next Objective

12 . 4 . We now proceed to the analysis of more complicated games.
The general one-person game having been disposed of, the simplest one
of the remaining games is the zero-sum two-person game. Accordingly
we are going to discuss it first.

Afterwards there is a choice of dealing either with the general two-
person game or with the zero sum three-person game. It will be seen that
our technique of discussion necessitates taking up the zero-sum three-person

1 We do not in the least intend, of course, to detract from the enormous importance
of those discoveries. It is just because of their great power that we are now in a position
to treat this side of the matter as briefly as we do. We are interested in those aspects of
the problem which are not settled by the concept of probability alone; consequently
these and not the satisfactorily settled ones must occupy our attention.

2 Concerning the important connection between the use of mathematical expectation
and the concept of numerical utility, cf. 3.7. and the considerations which precede it.

8 Some games like Roulette are of an even more peculiar character. In Roulette the
mathematical expectation of the players is clearly negative. Thus the motives for
participating in that game cannot be understood if one identifies the monetary return
with utility.

88

ZERO-SUM TWO-PERSON GAMES: THEORY

game first. After that we shall extend the theory to the zero-sum n- person
game (for all n = 1, 2, 3, • • • ) and only subsequently to this will it be
found convenient to investigate the general n- person game.

13. Functional Calculus
13.1. Basic Definitions

13.1.1. Our next objective is — as stated in 12.4. — the exhaustive dis-
cussion of the zero-sum two-person games. In order to do this adequately,
it will be necessary to use the symbolism of the functional calculus — or at
least of certain parts of it — more extensively than we have done thus far.
The concepts which we need are those of functions , of variables, of maxima
and minima, and of the use of the two latter as functional operations. All
this necessitates a certain amount of explanation and illustration, which
will be given here.

After that is done, we will prove some theorems concerning maxima,
minima, and a certain combination of these two, the saddle value. These
theorems will play an important part in the theory of the zero-sum two-
person games.

13.1.2. A function </> is a dependence which states how certain entities
x, y, • • • — called the variables of <f > — determine an entity u — called the
value of <t>. Thus u is determined by <f> and by the x, y, • • • , and this
determination — i.e. dependence — will be indicated by the symbolic equation

u = <t>(x, y, • • • ).

In principle it is necessary to distinguish between the function <f> itself —
which is an abstract entity, embodying only the general dependence of
u = <t>(x, y, • • • ) on the x, y, ■ * * — and its value <t>(x, y, • • • ) for any
specific x, y, • • • . In practical mathematical usage, however, it is often
convenient to write <l>(x, y, • • • ) — but with x, y, • • • indeterminate —
instead of <f> (cf. the examples (c)-(e) below; (a), (b) are even worse, cf.
footnote 1 below).

In order to describe the function <t> it is of course necessary — among
other things — to specify the number of its variables x, y, • • • . Thus
there exist one-variable functions <t>(x), two-variable functions <t>(x, y), etc.

Some examples:

(a) The arithmetical operations x + 1 and x 2 are one-variable functions. 1

(b) The arithmetical operations of addition and of multiplication x + y
and xy, are two-variable functions. 1

(c) For any fixed k the $k(ir) of 9.2.4. is a one-variable function (of w).
But it can also be viewed as a two-variable function (of k, w).

(d) For any fixed k the 2*(/c, D K ) of (11 :A) in 11.1.3. is a two-variable
function (of k, D«). 2

(e) For any fixed k the 3C*(7 i, • • • , r„) of 11.2.3. is a n-variable function
(of n, * * • , r n ). 1

1 Although they do not appear in the above canonical forms <t>(x), <t>(x, y).

* We could also treat k in (d) and k in (e) like k in (c), i.e. as a variable.

FUNCTIONAL CALCULUS

89

13, 1.3. It is equally necessary, in order to describe a function 0 to
specify for which specific choices of its variables x, y> • • • the value
0(x, Vi * • * ) is defined at all. These choices — i.e. these combinations — of
x, y, • • • form the domain of 0.

The examples (a)-(e) show some of the many possibilities for the domains
of functions: They may consist of arithmetical or of analytical entities, as
well as of others. Indeed:

(a) We may consider the domain to consist of all integer numbers, —
or equally well of all real numbers.

(b) All pairs of either category of numbers used in (a), form the domain.

(c) The domain is the set ft of all objects ir which represent the plays
of the game T (cf. 9.1.1. and 9.2.4.).

(d) The domain consists of pairs of a positive integer k and a set D«.

(e) The domain consists of certain systems of positive integers.

A function 0 is an arithmetical function if its variables are positive
integers; it is a numerical function if its variables are real numbers; it is a
set-function if its variables are sets (as, e.g., D* in (d)).

For the moment we are mainly interested in arithmetical and numerical
functions.

We conclude this section by an observation which is a natural conse-
quence of our view of the concept of a function. This is, that the number
of variables, the domain, and the dependence of the value on the varia-
bles, constitute the function as such: i.e., if two functions 0, 0 have the
same variables x, y, • • • and the same domain, and if 0(x, y f • • • ) =
0(x, y, • • • ) throughout this domain, then 0,0 are identical in all respects. 1

13.2. The Operations Max and Min

13.2.1. Consider a function <t> which has real numbers for values

</>(*, »,•••)•

Assume first that 0 is a one-variable function. If its variable can be
chosen, say as x = x 0 so that 0(x o ) ^ 0(x') for all other choices x', then we
say that 0 has the maximum 0(x o) and assumes it at x = xo.

Observe that this maximum 0(x o ) is uniquely determined; i.e., the
maximum may be assumed at x = x 0 for several x 0 , but they must all fur-
nish the same value 0(x o). 2 We denote this value by Max 0(x), the maxi-
mum value of 0(x).

If we replace ^ by then the concept of 0’s minimumy 0(x o ), obtains,
and of xo where 0 assumes it. Again there may be several such x 0 , but they
must all furnish the same value 0(x o ). We denote this value by Min 0(x),
the minimum value of 0.

1 The concept of a function is closely allied to that of a set, and the above should be
viewed in parallel with the exposition of 8.2.

* Proof : Consider two such xo, say xj and xj'. Then 4>(xJ) £ ^(x® ) and ^(x?) £ ^(xj).
Hence 0(xj) * <£(x' 0 ').

90

ZERO-SUM TWO-PERSON GAMES: THEORY

Observe that there is no a 'priori guarantee that either Max <f>(x) or
Min 4>(x) exist. 1

If, however, the domain of <t> — over which the variable x may run —
consists only of a finite number of elements, then the existence of both
Max <t>(x) and Min <f>(x) is obvious. This will actually be the case for most
functions which we shall discuss. 2 For the remaining ones it will be a
consequence of their continuity together with the geometrical limitations of
their domains. 8 At any rate we are restricting our considerations to such
functions, for which Max and Min exist.

13.2.2. Let now <f> have any number of variables x, y f z y • • • . By sin-
gling out one of the variables, say x, and treating the others, y y z, • • * , as
constants, we can view <t>(x, y, z, • • • ) as a one- variable function, of the
variable x. Hence we may form Max <f>(x y y y z y • • * ), Min <t> (x y y y z y • • • )
as in 13.2.1., of course with respect to this x.

But since we could have done this equally well for any one of the other
variables y y z y • • • it becomes necessary to indicate that the operations
Max, Min were performed with respect to the variable x. We do this by
writing Max* 4>(x, y y z, • • • ), Min* (f>(x, y y z y • • • ) instead of the incom-
plete expressions Max <f> y Min <f>. Thus we can now apply to the function
<f>(x y Vj z i * * * ) any one of the operators Max*, Min*, Max„, Min„, Max*,
Min*, • • • . They are all distinct and our notation is unambiguous.

This notation is even advantageous for one variable functions, and we
will use it accordingly; i.e. we write Max* <t>(x) y Min* <t>(x) instead of the
Max <£(x), Min <f>(x) of 13.2.1.

Sometimes it will be convenient — or even necessary — to indicate the
domain S for a maximum or a minimum explicitly. E.g. when the func-
tion <t>(x) is defined also for (some) x outside of S y but it is desired to form the
maximum or minimum within S only. In such a case we write

Max* in5 <t>(x), Min* i n s <t>(x)
instead of Max* <t>(x) y Min* <t>(x).

In certain other cases it may be simpler to enumerate the values of 4>{x) —
say a, b y • • • — than to express <t>(x) as a function. We may then write

1 E.g. if <f>{x) ® x with all real numbers as domain, then neither Max <f>( x) nor Min
exist.

2 Typical examples: The functions 3Ck\ri y • • • , r„) of 11.2.3. (or of (e) in 13.1.2.), the
function 3C(ri, r 2 ) of 14.1.1.

8 Typical examples: The functions K( £ , 17 ), Max-> K( £ ,

17 ), Min-> K( | , 17 ) in
v

fix fit

17.4., the functions Min Tj ^ 3C(n, t 2 )£ v Max fi 2) ^( r i» T *)v r in 17.5.2. The vari-

r i *“ 1 U- 1

— ► — >

ables of all these functions are £ or 17 or both, with respect to which subsequent maxima
and minima are formed.

Another instance is discussed in 46.2.1. espec. footnote 1 on p. 384, where the
mathematical background of this subject and its literature are considered. It seems
unnecessary to enter upon these here, since the above examples are entirely elementary.

FUNCTIONAL CALCULUS 91

Max (a, &,-••)> [Min (a, 6, • • • )] instead of Max x <£(x), [Min*

13 . 2 . 3 . Observe that while <f>(x, y, z, • • • ) is a function of the variables
x, y, z, • • • , Max* <£(x, y, z, • • • ), Min* </>(x, $/, z, • • • ) are still func-
tions, but of the variables 2 /, z, • • • only. Purely typographically, x is
still present in Max* <£(x, y, z, • • • ), Min* <£(x, y, z, • • • ), but it is no
longer a variable of these functions. We say that the operations Max*,
Min* kill the variable x which appears as their index. 2

Since Max* <£(x, y, z, • • • ), Min* <f>(x, y } z, • • • ) are still functions

of the variables y } z, • • • , 3 we can go on and form the expressions

Maxy Max* <£(x, y } z, • • • ), Max y Min* <£(x, 2/> z, • • • ),

Miny Max* 4>(x, t/, z, • • • ), Min y Min* </>(x, t/, z, • • • ),

We could equally form

Max* Maxy <£(x, y, z, • • • ), Max* Miny <£(x, t/, z, • • • )

etc. ; 4 or use two other variables than x, y (if there are any) ; or use more
variables than two (if there are any).

Infiney after having applied as many operations Max or Min as there are
variables of <£(x, y, z, • • • ) — in any order and combination, but precisely
one for each variable x y y t z, • • * — we obtain a function of no variables
at all, i.e. a constant.

13.3. Commutativity Questions

13.3.1. The discussions of 13.2.3. provide the basis for viewing the Max*,
Min*, Maxy, Min y , Max,, Min,, • • • entirely as functional operations , each
one of which carries a function into another function. 6 We have seen that
we can apply several of them successively. In this latter case it is prima
fade relevant, in which order the successive operations are applied.

But is it really relevant? Precisely: Two operations are said to commute
if, in case of their successive application (to the same object), the order in
which this is done does not matter. Now we ask: Do the operations Max*,
Min*, Maxy, Min y , Max,, Min,, - - - all commute with each other or not?

We shall answer this question. For this purpose we need use only two
variables, say x, y and then it is not necessary that <p be a function of further
variables besides x, y. 6

1 Of course Max (a, b, • • • ) [Min (a, b } • • • )] is simply the greatest [smallest] one
among the numbers a, 6, • • •

* A well known operation in analysis which kills a variable x is the definite integral:
4>(x) is a function of x, but <f>(x)dx is a constant.

8 We treated z, ■ • • as constant parameters in 13.2.2. But now that x has been
killed we release the variables ?/, z, • • • .

4 Observe that if two or more operations are applied, the innermost applies first and
kills its variable; then the next one follows suit, etc

6 With one variable less, since these operations kill one variable each.

8 Further variables of <t > , if any, may be treated as constants for the purpose of this
analysis.

92

ZERO-SUM TWO-PERSON GAMES: THEORY

So we consider a two- variable function </>(x, y). The significant ques-
tions of commutativity are then clearly these:

Which of the three equations which follow are generally true:

(13:1) Max, Max* 4>{x, y) = Max y Max, tf>(x, y),

(13:2) Min, Min y <t>(x, y) = Min y Min, <t>(x , y),

(13:3) Max, Min y <£(x, y) = Min y Max, <t>(x, y). 1

We shall see that (13:1), (13:2) are true, while (13:3) is not; i.e., any two
Max or any two Min commute, while a Max and a Min do not commute in
general. We shall also obtain a criterion which determines in which special
cases Max and Min commute.

This question of commutativity of Max and Min will turn out to be
decisive for the zero-sum two-person game (cf. 14.4.2. and 17.6.).

13.3.2. Let us first consider (13:1). It ought to be intuitively clear that
Max, Max y <f>(x , y) is the maximum of <t>(x , y) if we treat x, y together as one
variable; i.e. that for some suitable x ( >, y 0 , </>(x 0 , Vo) — Max, Max y 0(x, y)
and that for all x', y', </>(x 0 , yo) S </>(x', y').

If a mathematical proof is nevertheless wanted, we give it here: Choose
x 0 so that Max y <£(x, y) assumes its x-maximum at x = x 0 , and then choose y 0
so that </>(x o, y) assumes its y-maximum at y = y 0 . Then

<t>(x o, y 0 ) = Max y 4>(x 0 , y) = Max, Max y <t>(x y y),
and for all x', y'

<t>(x o, yo) = Max y <p(xo, y) ^ Max y <t>(x', y) ^ </>(x', y').

This completes the proof.

Now by interchanging x, y we see that Max y Max, </>(x, y) is equally
the maximum of <t>(x, y) if we treat x, y as one variable.

Thus both sides of (13:1) have the same characteristic property, and
therefore they are equal to each other. This proves (13:1).

Literally the same arguments apply to Mm in place of Max: we need
only use ^ consistently in place of ^ . This proves (13:2).

This device of treating two variables x, y as one, is occasionally quite
convenient in itself. When we use it (as, e.g., in 18.2.1., with r i, 7% 3 C(ti, t 2 )
in place of our present x, y, <t>(x, y)), we shall write Max,, y <£(x, y) and
Min,. y <£(x, y).

13.3.3. At this point a graphical illustration may be useful. Assume
that the domain of <f> for x, y is a finite set. Denote, for the sake of sim-
plicity, the possible values of x (in this domain) by 1, • • • , t and those of
y by 1, * * * , 8. Then the values of </>(x, y) corresponding to all x, y in this
domain — i.e. to all combinations of x = 1, • • • , /, y = 1, • • • , s — can
be arranged in a rectangular scheme: We use a rectangle of t rows and s

1 The combination Min, Max y requires no treatment of its own, since it obtains from
the above — Max, Min y — by interchanging x } y.

FUNCTIONAL CALCULUS

93

columns, using the number x = 1, • • • , t to enumerate the former and
the number y = 1, • • • , s to enumerate the latter. Into the field of inter-
section of row x and column y — to be known briefly as the field x , y — we
write the value <t>(x, y) (Fig. 11). This arrangement, known in mathematics
as a rectangular matrix , amounts to a complete characterization of the func-
tion <t>(x, y ). The specific values <t>(x, y) are the matrix elements .

1

2

y

8

1

♦u, i)

♦(1, 2)

*( 1 , V )

^(1, •)

2

*(2, 1)

*(2, 2)

*( 2 , V )

*(2, «)

*

X

<t>(x, 1 )

4>(x, 2)

y)

1

»)

•

t

1)

<p(l, 2)

y)

*)

Figure 11.

Now Max y <t>(x, y) is the maximum of </>(#, y) in the row x.

Max z Maxy 0(x, y)

is therefore the maximum of the row maxima. On the other hand,

Max* y)

is the maximum of 4>{x } y) in the column y. Max y Max* <t>(x y y) is therefore
the maximum of the column maxima. Our assertions in 13.3.2. concerning
(13:1) can now be stated thus: The maximum of the row maxima is the
same as the maximum of the column maxima ; both are the absolute maxi-
mum of 4>{x , y) in the matrix. In this form, at least, these assertions should
be intuitively obvious. The assertions concerning (13:2) obtain in the
same way if Min is put in place of Max.

13.4. The Mixed Case. Saddle Points

13 . 4 . 1 . Let us now consider (13:3). Using the terminology of 13.3.3.
the left-hand side of (13:3) is the maximum of the row minima and the
right-hand side is the minimum of the column maxima. These two numbers
are neither absolute maxima, nor absolute minima, and there is no prima
facie evidence why they should be generally equal. Indeed, they are not.
Two functions for which they are different are given in Figs. 12, 13. A

94

ZERO-SUM TWO-PERSON GAMES: THEORY

function for which they are equal is given in Fig. 14. (All these figures
should be read in the sense of the explanations of 13.3.3. and of Fig. 11.)

These figures — as well as the entire question of commutativity of Max
and Min — will play an essential role in the theory of zero-sum two-person
games. Indeed, it will be seen that they represent certain games which
are typical for some important possibilities in that theory (cf. 18.1.2.).
But for the moment we want to discuss them on their own account, without
any reference to those applications.

<== 8=2

1

2

row

minima

1

1

-1

-1

2

-1

1

-1

column

maxima

1

1

Maximum of row minima = — 1
Minimum of column maxima = 1
Figure 12.

<=- 8=3

1

2

3

row

minima

1

0

-1

1

-1

2

1

0

-1

-1

3

-1

1

0

-1

column

maxima

1

1

1

1

Maximum of row minima = — 1
Minimum of column maxima = 1
Figure 13.

<=8 = 2

1

2

row

minima

1

-2

1

-2

2

-1

2

-1

column

maxima

-1

2

Maximum of row minima = — 1
Minimum of column maxima = — 1
Figure 14.

13 . 4 . 2 . Since (13:3) is neither generally true, nor generally false, it is
desirable to discuss the relationship of its two sides

(13:4) Max* Min y <p(x ) y ), Min y Max* <f>(x y y ),

more fully. Figs. 12-14, which illustrated the behavior of (13:3) to a
certain degree, give some clues as to what this behavior is likely to be.

Specifically:

(13 :A) In all three figures the left-hand side of (13:3) (i.e. the first

expression of (13:4)) is ^ the right-hand side of (13:3) (i.e. the
second expression in (13:4)).

FUNCTIONAL CALCULUS

95

(13 :B) In Fig. 14 — where (13:3) is true — there exists a field in the

matrix which contains simultaneously the minimum of its row
and the maximum of its column. (This happens to be the ele-
ment — 1 — the left lower corner field of the matrix.) In the
other figures 12, 13, where (13:3) is not true, there exists no
such field.

It is appropriate to introduce a general concept which describes the
behavior of the field mentioned in (13 :B). We define accordingly:

Let </>(x, y) be any two-variable function. Then x 0 , yo is a saddle point
of <f> if at the same time </>(x, y 0 ) assumes its maximum at x = x 0 and <£(x 0 , y)
assumes its minimum at y = y 0 .

The reason for the use of the name saddle point is this: Imagine the
matrix of all x, y elements (x = 1, • * • , t } y = 1, • • * s; cf. Fig. 11)
as an oreographical map, the height of the mountain over the field x, y
being the value of </>(x, y) there. Then the definition of a saddle point
x 0 , 2/o describes indeed essentially a saddle or pass at that point (i.e. over the
field x 0 , y o) ; the row x 0 is the ridge of the mountain, and the column y 0 is
the road (from valley to valley) which crosses this ridge.

The formula (13:C*) in 13.5.2. also falls in with this interpretation. 1

13 . 4 . 3 . Figs. 12, 13 show that a <t> may have no saddle points at all.
On the other hand it is conceivable that possesses several saddle points.
But all saddle points x 0 , yo — if they exist at all — must furnish the same
value </>(x o, yo)- 2 We denote this value — if it exists at all — by Sa x/v </>(x, y) t
the saddle value of </>(x, y). 3

We now formulate the theorems which generalize the indications of
(13 :A), (13 :B). We denote them by (13 :A*), (13 :B*), and emphasize
that they are valid for all functions <£(x, y).

(13:A*) Always

Max* Min y </>(x, y) ^ Min y Max* </>(x, y).

(13 :B*) We have

Max* Min y <f>(x, y) = Min y Max* </>(x, y)
if and only if a saddle point x 0 , yo of <t> exists.

13.5. Proofs of the Main Facts

13 . 5 . 1 . We define first two sets A+, B+ for every function </>(x, y ).
Min y <£(x, y) is a function of x; let A* be the set of all those x 0 for which

1 All this is closely connected with — although not precisely a special case of—-certain
more general mathematical theories involving extremal problems, calculus of variations,
etc. Cf. M. Morse: The Critical Points of Functions and the Calculus of Variations in
the Large, Bull. Am. Math. Society, Jan .-Feb. 1929, pp. 38 cont., and What is Analysis in
the Large?, Am. Math. Monthly, Vol. XLIX, 1942, pp. 358 cont.

1 This follows from (13 :C*) in 13.5.2. There exists an equally simple direct proof:
Consider two saddle points x 0 , yo, say xj, yj and x' 0 ', y". Then:

, yo) - Max* <*>(x, y' 0 ) £ <*>(*", yj) £ Min y *(x' 0 ', y) - <*>(x' 0 ', y'o')»

i.e.: <HxJ, y'o) £ Vo)- Similarly 4>{x ' 0 ', y' 0 ') £ <*>(xj, yo).

Hence *(xj, yj) » 0(x o ', y' 0 ' ).

* Clearly the operation Sa*/ y tf>(x, y) kills both variables x, y. Cf. 13.2.3.

96

ZERO-SUM TWO-PERSON GAMES: THEORY

this function assumes its maximum at x = Xo. Max* <t>(x , y ) is a function
of y ; let B+ be the set of all those yo for which this function assumes its mini-
mum at y = yo-

We now prove (13:A*), (13:B*).

Proof of (13 :A*): Choose xo in A* and y 0 in B+. Then

Max* Min y <t>(x, y) = Min y <j>(x 0y y) g <t>(xo, yo)

S Max* <f>(x y yo) = Min y Max* <t>(x y y) y

i.e.: Max* Min y <f>(x , y) ^ Min y Max* 4>(x, y) as desired.

Proof of the necessity of the existence of a saddle point in (13:B*):
Assume that

Max* Min y <f>(x, y) = Min y Max* <f>(x y y).

Choose x 0 in A* and y 0 in B+\ then we have

Max* <t>(x , y 0 ) = Min y Max* </>(x, y)

= Max* Min y <f>(x y y) = Min y </>(x 0 , y).

Hence for every x'

<t>(x' y y 0 ) g Max* <f>(x y y 0 ) = Min y <t>(x 0 , y) ^ <t>(x 0 , yo ),

i.e. <f>(x o, yo) ^ </>(x', y 0 ) — so </>(x, y 0 ) assumes its maximum at x = x 0 .
And for every y f

4>{x o, y f ) ^ Min y <f>(x 0y y) = Max* tf>(x, yo) ^ <t>(x 0y y Q ) y

i.e. <t>(x o, yo) ^ 2/') — so </>(xo, 2/) assumes its minimum at y = 2/o-

Consequently these x 0 , 2/o form a saddle point.

Proof of the sufficiency of the existence of a saddle point in (13 :B*):
Let xo, 2/o be a saddle point. Then

Max* Min y <j>(x y y) ^ Min y <t>(x Qy y) = <t>(x 0y yo),

Min y Max* 4>(x y y) g Max* <t>(x y y 0 ) = <t>(x 0 , yo),

hence

Max* Min y 4>(x y y) ^ <f>(x 0y y Q ) ^ Min y Max* <t>(x y y).

Combining this with (13 :A*) gives

Max* Min y <j>{x y y) = <f>(x 0 , yo) = Min y Max* <t>(x y y) y
and hence the desired equation.

13 . 6 . 2 . The considerations of 13.5.1. yield some further results which
are worth noting. We assume now the existence of saddle points, — i.e.
the validity of the equation of (13 :B*).

For every saddle point x 0 , yo

(13:C*) 0(x o, yo) = Max* Min y 4>{x y y) = Min y Max* <f>(x y y).

Proof: This coincides with the last equation of the sufficiency proof of
(13:B*) in 13.5.1.

FUNCTIONAL CALCULUS

97

(13 :D*) x 0 , 2/0 is a saddle point if and only if xo belongs to A ♦ and |/o

belongs to B+. 1

Proof of sufficiency: Let x 0 belong to A+ and 2/0 belong to B+. Then the
necessity proof of (13:B*) in 13.5.1. shows exactly that this x 0 , 2/0 is a saddle
point.

Proof of necessity: Let x 0 , yo be a saddle point. We use (13 :C*). For
every x f

Min„ 0(x', y) £ Max* Min y <f>(x , 2/) = </>(zo, 2/o) = Min y <j>(x 0} y) }

i.e. Min y </>(x 0 , 2/) ^ Min y </>(x', 1/) — so Min y </>(x, y) assumes its maximum
at x = x 0 . Hence x 0 belongs to A*. Similarly for every y f

Max* <t>(x f y') ^ Min y Max* <f>(x, y) = <t>(x 0 , y 0 ) = Max* 4>{x , 2/0),

i.e. Max* </>(x, 2/0) ^ Max* <t>(x, y f ) — so Max* <£(x, 2/) assumes its minimum
at y = 2/0. Hence 2/ belongs to 5*. This completes the proof.

The theorems (13:C*), (13:D*) indicate, by the way, the limitations
of the analogy described at the end of 13.4.2.; i.e. they show that our con-
cept of a saddle point is narrower than the everyday (oreographical) idea
of a saddle or a pass. Indeed, (13 :C*) states that all saddles — provided
that they exist — are at the same altitude. And (13 :D*) states — if we depict
the sets A+, B* as two intervals of numbers 2 — that all saddles together are
an area which has the form of a rectangular plateau. 8

13.5.3. We conclude this section by proving the existence of a saddle
point for a special kind of x, y and y). This special case will be seen
to be of a not inconsiderable generality. Let a function ^(x, u) of two
variables x, u be given. We consider all functions /(x) of the variable
which have values in the domain of u. Now we keep the variable x but
in place of the variable u we use the function / itself. 4 The expression
’/'(•£>/(£)) is determined by x,/; hence we may treat ^(x,/(x)) as a function of
the variables x, / and let it take the place of <f>(x, y).

We wish to prove that for these x, / and ^(x, /(x)) — in place of x, y
and </>(x, y) — a saddle point exists; i.e. that

(13 :E) Max* Min/ ^(x, /(x)) = Min/ Max* ^(x, /(x)).

Proof: For every x choose a Uq with ^(x, u 0 ) = Min u ^(x, u). This Uo
depends on x, hence we can define a function / 0 by Uq = /o(x). Thus
\f/(x, / 0 (x)) = Min* \P(x, u). Consequently

Max* ^(x, fo(x)) = Max* Min u ^(x, u ).

1 Only under our hypothesis at the beginning of this section! Otherwise there exist
no saddle points at all.

1 If the x, y are positive integers, then this can certainly be brought about by two
appropriate permutations of their domain.

3 The general mathematical concepts alluded to in footnote 1 on p. 95 are free from
these limitations. They correspond precisely to the everyday idea of a pass.

4 The reader is asked to visualiae this: Although itself a function, / may perfectly well
be the variable of another function.

98 ZERO-SUM TWO-PERSON GAMES: THEORY

A fortiori ,

(13 :F) Min / Max* ^(x, f(x)) ^ Max* Min u ^(x, u).

Now Min/ \p(x, f(x)) is the same thing as Min u \p(x , w) since / enters into
this expression only via its value at the one place x, i.e. /(x), for which we
may write u . So Min/ \p(x, f(x)) = Min u ^(x, u) and consequently,

(13 :G) Max* Min/ \p(x, /(x)) = Max* Min u ^(x, w).

(13 :F), (13:G) together establish the validity of a ^ in (13:E). The ^
in (13 :E) holds owing to (13 :A*). Hence we have = in (13 :E), i.e. the
proof is completed.

14. Strictly Determined Games
14.1. Formulation of the Problem

14.1.1. We now proceed to the consideration of the zero-sum two-person
game. Again we begin by using the normalized form.

According to this the game consists of two moves: Player 1 chooses a
number ri = 1, • • • , player 2 chooses a number r 2 = 1, • • • , 0 2 , each
choice being made in complete ignorance of the other, and then players
1 and 2 get the amounts 3 Ci(ti, t 2 ) and 3C 2 (ti, t 2 ), respectively. 1

Since the game is zero-sum, we have, by 11.4.

3Ci(t i, t 2 ) + 3C*(ri, t 2 ) as 0.

We prefer to express this by writing

3Ci(ti, t 2 ) = 3C(ti, r 2 ), 3C 2 (ri, r 2 ) = — JC(ri, r 2 ).

We shall now attempt to understand how the obvious desires of the
players 1, 2 will determine the events, i.e. the choices n, r 2 .

It must again be remembered, of course, that n, r 2 stand ultima analysi
not for a choice (in a move) but for the players' strategies; i.e. their entire
“theory” or “plan” concerning the game.

For the moment we leave it at that. Subsequently we shall also go
“behind” the n, r 2 and analyze the course of a play.

14.1.2. The desires of the players 1, 2, are simple enough. 1 wishes
to make 3 Ci(ti, r 2 ) = X(r lf t 2 ) a maximum, 2 wishes to make 3C 2 (ti, r 2 ) ss
— JC (ri, r 2 ) a maximum; i.e. 1 wants to maximize and 2 wants to minimize
3C(ri, r 2 ).

So the interests of the two players concentrate on the same object: the
one function 3C(ri, t 2 ). But their intentions are — as is to be expected in a
zero-sum two-person game — exactly opposite : 1 wants to maximize, 2 wants
to minimize. Now the peculiar difficulty in all this is that neither player
has full control of the object of his endeavor — of X(r h r 2 ) — i.e. of both
its variables r i, r 2 . 1 wants to maximize, but he controls only t x \ 2 wants to

minimize, but he controls only r 2 : What is going to happen?

*Cf. (1 1 :D) in 11.2.3.

STRICTLY DETERMINED GAMES

99

The difficulty is that no particular choice of, say r i, need in itself make
3C(ri, t 2 ) either great or small. The influence of r x on 3C(ri, t 2 ) is, in general,
no definite thing; it becomes that only in conjunction with the choice of
the other variable, in this case r 2 . (Cf. the corresponding difficulty in
economics as discussed in 2.2.3.)

Observe that from the point of view of the player 1 who chooses a
variable, say n, the other variable can certainly not be considered as a
chance event. The other variable, in this case r 2 , is dependent upon the
will of the other player, which must be regarded in the same light of “ ration-
ality ” as one’s own. (Cf. also the end of 2.2.3. and 2.2.4.)

14 . 1 . 3 . At this point it is convenient to make use of the graphical
representation developed in 13.3.3. We represent 3C(ri, t 2 ) by a rectangular
matrix: We form a rectangle of 0i rows and 0 2 columns, using the number
n = 1 , • • • , #1 to enumerate the former, and the number r 2 = 1 , • • • ,0 2
to enumerate the latter; and into the field n, r 2 we write the matrix element
5C(ri, t 2 ). (Cf. with Figure 11 in 13.3.3. The <f>, x, y, t, s there correspond
to our 3C, n, r 2 , p h 0 2 (Figure 15).)

1

2

; •

r 2

0*

1

3C(1, 1)

3C(1, 2)

3C(1, r 2 )

3C(1, ft)

2

0C(2, 1)

3C(2, 2)

3C(2, r,)

3C(2, 0j)

r i

3C(r„ 1)

3C(r,, 2)

3C(n, r*)

3C(r,, ft)

0i

«C(ft, 1)

3C(0,, 2)

3C(ft, ft)

Figure 15.

It ought to be understood that the function X(ti, r 2 ) is subject to no
restrictions whatsoever; i.e., we are free to choose it absolutely at will. 1
Indeed, any given function 3C(ri, r 2 ) defines a zero-sum two-person game
in the sense of (11 :D) of 11.2.3. by simply defining

3Ci(ti, t 2 ) = 3C(ti, t 2 ), 3C 2 (ti, r 2 ) ss — 3C(ri, r 2 )

(cf. 14.1.1.). The desires of the players 1, 2, as described above in the
last section, can now be visualized as follows: Both players are solely

1 The domain, of course, is prescribed: It consists of all pairs n, r% with n — 1, • • • ,
0i ; T t « 1, • • ■ , 0*. This is a finite set, so all Max and Min exist, cf. the end of 13.2.1.

100

ZERO-SUM TWO-PERSON GAMES: THEORY

interested in the value of the matrix element JC(ri, r 2 ). Player 1 tries to
maximize it, but he controls only the row, — i.e. the number n. Player 2
tries to minimize it, but he controls only the column, — i.e. the number r 2 .

We must now attempt to find a satisfactory interpretation for the out-
come of this peculiar tug-of-war. 1

14.2. The Minorant and the Majorant Games

14 . 2 . Instead of attempting a direct attack on the game T itself — for
which we are not yet prepared — let us consider two other games, which are
closely connected with T and the discussion of which is immediately feasible.

The difficulty in analyzing r is clearly that the player 1, in choosing n
does not know what choice r 2 of the player 2 he is going to face and vice
versa . Let us therefore compare T with other games where this difficulty
does not arise.

We define first a game Ti, which agrees with r in every detail except that
player 1 has to make his choice of r i before player 2 makes his choice of
r 2 , and that player 2 makes his choice in full knowledge of the value given
by player 1 to r i (i.e. Ts move is preliminary to 2’s move). 2 In this game
Ti player 1 is obviously at a disadvantage as compared to his position
in the original game T. We shall therefore call T x the minorant game of T.

We define similarly a second game r 2 which again agrees with T in every
detail except that now player 2 has to make his choice of r 2 before player 1
makes his choice of r x and that 1 makes his choice in full knowledge
of the value given by 2 to r 2 (i.e. 2's move is preliminary to Vs move). 3 In
this game r 2 the player 1 is obviously at an advantage as compared to
his position in the game T. We shall therefore call r 2 the majorant game
of r.

The introduction of these two games r i, r 2 achieves this: It ought to
be evident by common sense — and we shall also establish it by an exact
discussion — that for T h r 2 the “best way of playing” — i.e. the concept of
rational behavior — has a clear meaning. On the other hand, the game T
lies clearly “between” the two games T h r 2 ; e.g. from Vs point of view T x
is always less and r 2 is always more advantageous than T. 4 Thus Ti, r 2
may be expected to provide lower and upper bounds for the significant
quantities concerning T. We shall, of course, discuss all this in an entirely
precise form. A priori , these “bounds” could differ widely and leave a
considerable uncertainty as to the understanding of T. Indeed, prima facie
this will seem to be the case for many games. But we shall succeed in
manipulating this technique in such a way — by the introduction of certain

1 The point is, of course, that this is not a tug-of-war. The two players have opposite
interests, but the means by which they have to promote them are not in opposition to
each other. On the contrary, these “means” — i.e. the choices of n, rj — are apparently
independent. This discrepancy characterizes the entire problem.

* Thus T \ — while extremely simple — is no longer in the normalized form.

* Thus Tj — while extremely simple — is no longer in the normalized form.

4 Of course, to be precise we should say “less than or equal to” instead of “less,” and
“more than or equal to” instead of “more.”

STRICTLY DETERMINED GAMES

101

further devices — as to obtain in the end a precise theory of T, which gives
complete answers to all questions.

14.3. Discussion of the Auxiliary Games

14 . 3 . 1 . Let us first consider the minorant game T i. After player 1
has made his choice ri the player 2 makes his choice r 2 in full knowledge
of the value of n. Since 2’s desire is to minimize 3C(ri, t 2 ), it is certain that
he will choose r 2 so as to make the value of 3C(ri, r 2 ) a minimum for this r 1 .
In other words: When 1 chooses a particular value of r 1 he can already foresee
with certainty what the value of 3C(ri, r 2 ) will be. This will be Min rj 0C(ri, t 2 ). 1
This is a function of n alone. Now 1 wishes to maximize 3C(ri, r 2 ) and
since his choice of ri is conducive to the value Min Tj 3C(r h r 2 ) — which depends
on r 1 only, and not at all on r 2 — so he will choose r 1 so as to maximize
Min Tj JC(ri, t 2 ). Thus the value of this quantity will finally be

Max Ti Min Tj 3C(r h r 2 ). 2

Summing up:

(14:A:a) The good way (strategy) for 1 to play the minorant game

Ti is to choose r 1 , belonging to the set A , — A being the set of
those ri for which Min Tj 3C(ri, r 2 ) assumes its maximum value
Max Tj Min fj 3C(ri, r 2 ).

(14:A:b) The good way (strategy) for 2 to play is this: If 1 has

chosen a definite value of n, 8 then r 2 should be chosen belong-
ing to the set B Tx , — B Tx being the set of those r 2 for which
3C(ri, t 2 ) assumes its minimum value Min Tj 3 C(t l , r 2 ). 4

On the basis of this we can state further:

(14:A:c) If both players 1 and 2 play the minorant game T j well,

i.e. if ri belongs to A and r 2 belongs to B Tx then the value of
3C(ri, r 2 ) will be equal to

Vi = Max Ti Min Tj 3 C(ti, r 2 ).

1 Observe that r 2 may not be uniquely determined: For a given n the r 2 -function
3C(n, r 2 ) may assume its r 2 -minimum for several values of r 2 . The value of 3C(n, r 2 )
will, however, be the same for all these r 2 , namely the uniquely defined minimum value
Min r ,3C(n, r 2 ). (Cf. 13.2.1.)

* For the same reason as in footnote 1 above, the value of n may not be unique, but the
value of Min Tf 3C(ri, r 2 ) is the same for all n in question, namely the uniquely-defined
maximum value

Max Tl Min Tj 3C(n r 2 ).

8 2 is informed of the value of n when called upon to make his choice of r 2 , — this is
the rule of Tu It follows from our concept of a strategy (cf. 4.1.2. and end of 11.1.1.)
that at this point a rule must be provided for 2’s choice of r 2 for every value of n, — irre-
spective of whether 1 has played well or not, i.e. whether or not the value chosen belongs
to A.

4 In all, this n is treated as a known parameter on which everything depends, — includ-
ing the set B Ti from which r 2 ought to be chosen.

102 ZERO-SUM TWO-PERSON GAMES: THEORY

The truth of the above assertion is immediately established in the mathe-
matical sense by remembering the definitions of the sets A and B v and by
substituting accordingly in the assertion. We leave this exercise — which
is nothing but the classical operation of “ substituting the defining for the
defined” — to the reader. Moreover, the statement ought to be clear by
common sense.

The entire discussion should make it clear that every play of the game
Ti has a definite value for each player. This value is the above Vi for the
player 1 and therefore — Vi for the player 2.

An even more detailed idea of the significance of Vi is obtained in this
way:

(14:A:d) Player 1 can, by playing appropriately, secure for himself

a gain ^ Vi irrespective of what player 2 does. Player 2
can, by playing appropriately, secure for himself a gain ^ — Vi,
irrespective of what player 1 does.

{Proof: The former obtains by any choice of t\ in A. The latter obtains
by any choice of r 2 in B r A Again we leave the details to the reader; they
are altogether trivial.)

The above can be stated equivalently thus:

(14:A:e) Player 2 can, by playing appropriately, make it sure that

the gain of player 1 is ^ Vi, i.e. prevent him from gaining
> Vi irrespective of what player 1 does. Player 1 can, by play-
ing appropriately, make it sure that the gain of player 2 is
^ — Vi, i.e. prevent him from gaining > — Vi irrespective of
what player 2 does.

14.3.2. We have carried out the discussion of r i in rather profuse detail
although the “ solution” is a rather obvious one. That is, it is very likely
that anybody with a clear vision of the situation will easily reach the same
conclusions “unmathematically,” just by exercise of common sense.
Nevertheless we felt it necessary to discuss this case so minutely because
it is a prototype of several others to follow where the situation will be much
less open to “ unmathematical ” vision. Also, all essential elements of
complication as well as the bases for overcoming them are really present
in this very simplest case. By seeing their respective positions clearly
in this case, it will be possible to visualize them in the subsequent, more
complicated, ones. And it will be possible, in this way only, to judge
precisely how much can be achieved by every particular measure.

14.3.3. Let us now consider the majorant game r 2 .

r 2 differs from T i only in that the roles of players 1 and 2 are inter-
changed: Now player 2 must make his choice r 2 first and then the player 1
makes his choice of r i in full knowledge of the value of r 2 .

1 Recall that n must be chosen without knowledge of r 2 , while r 2 is chosen in full
knowledge of n.

STRICTLY DETERMINED GAMES

103

But in saying that r 2 arises from Ti by interchanging the players 1 and 2,
it must be remembered that these players conserve in the process their
respective functions 0Ci(ri, r 2 ), 3C 2 (tj, r 2 ), i.e. 3C(ri, t 2 ), — 3C(ti, t 2 ). That
is, 1 still desires to maximize and 2 still desires to minimize 3 C(ti, r 2 ).

These being understood, we can leave the practically literal repetition
of the considerations of 14.3.1. to the reader. We confine ourselves to
restating the significant definitions, in the form in which they apply to r 2 .

(14:B:a) The good way (strategy) for 2 to play the majorant game

r 2 is to choose t 2 belonging to the set B , — B being the set of
those t 2 for which Max Tj 3C(ri, t 2 ) assumes its minimum value
Min Tj Max Tj 3C(ri, t 2 ).

(14:B:b) The good way (strategy) for 1 to play is this: If 2 has

chosen a definite value of r 2 , 1 then t\ should be chosen belong-
ing to the set A Tj , — A Tx being the set of those r i for which
3 C(ti, r 2 ) assumes its maximum value Max Tj 3C (r h r 2 ). 2

On the basis of this we can state further:

(14:B:c) If both players 1 and 2 play the majorant game r 2 well,

i.e. if r 2 belongs to B and ri belongs to A Tt then the value of
3C(ri, r 2 ) will be equal to

v 2 = Min Tj Max Ti 3C(ri, r 2 ).

The entire discussion should make it clear that every play of the game
r 2 has a definite value for each player. This value is the above v 2 for the
player 1 and therefore — v 2 for the player 2.

In order to stress the symmetry of the entire arrangement, we repeat,
mutatis mutandis , the considerations which concluded 14.3.1. They now
serve to give a more detailed idea of the significance of v 2 .

(14:B:d) Player 1 can, by playing appropriately, secure for himself

a gain ^ v 2 , irrespective of what player 2 does. Player 2
can, by playing appropriately, secure for himself a gain
— v 2 , irrespective of what player 1 does.

(Proof: The latter obtains by any choice of r 2 in B. The former obtains
by any choice of t\ in A r 2 . 8 Cf. with the proof, loc. cit.)

The above can again be stated equivalently thus :

(14:B:e) Player 2 can, by playing appropriately, make it sure

that the gain of player 1 is g v 2 , i.e. prevent him from gaining

1 1 is informed of the value of r% when called upon to make his choice of n — this is the
rule of r a (Cf. footnote 3 on p. 101).

2 In all this r % is treated as a known parameter on which everything depends, including
the set Ar t from which n ought to be chosen.

* Remember that r% must be chosen without any knowledge of n, while n is chosen
with full knowledge of n.

104

ZERO-SUM TWO-PERSON GAMES: THEORY

> v 2 , irrespective of what player 1 does. Player 1 can, by
playing appropriately, make it sure that the gain of player 2
is ^ — v 2 , i.e. prevent him from gaining > — v 2 , irrespective
of what player 2 does.

14.3.4. The discussions of Ti and r 2 , as given in 14.3.1. and 14.3.3.,
respectively, are in a relationship of symmetry or duality to each other;
they obtain from each other, as was pointed out previously (at the begin-
ning of 14.3.3.) by interchanging the roles of the players 1 and 2. In itself
neither game Ti nor r 2 is symmetric with respect to this interchange; indeed,
this is nothing but a restatement of the fact that the interchange of the
players 1 and 2 also interchanges the two games Ti and r 2 , and so modifies
both. It is in harmony with this that the various statements which we
made in 14.3.1. and 14.3.3. concerning the good strategies of Ti and r 2 ,
respectively — i.e. (14:A:a), (l4:A:b), (14:B:a), (14 :B :b), loc. cit. — were
not symmetric with respect to the players 1 and 2 either. Again we see:
An interchange of the players 1 and 2 interchanges the pertinent definitions
for Ti and r 2 , and so modifies both. 1

It is therefore very remarkable that the characterization of the value
of a play (vi for r i, v 2 for r 2 ), as given at the end of 14.3.1. and 14.3.3. — i.e.
(14:A:c), (14:A:d), (14:A:e), (14:B:c), (14:B:d), (14:B:e), loc. cit. (except
for the formulae at the end of (14:A:c) and of (14:B:c)) — are fully
symmetric with respect to the players 1 and 2. According to what was
said above, this is the same thing as asserting that these characterizations
are stated exactly the same way for I\ and r 2 . 2 All this is, of course, equally
clear by immediate inspection of the relevant passages.

Thus we have succeeded in defining the value of a play in the same way
for the games Ti and r 2 , and symmetrically for the players 1 and 2: in
(14:A:c), (14:A:d), (14:A:e), (14:B:c), (14:B:d) and (14:B:e) in 14.3.1.
and in 14.3.3., — this in spite of the fundamental difference of the individual
role of each player in these two games. From this we derive the hope
that the definition of the value of a play may be used in the same form for
other games as well — in particular for the game r — which, as we know,
occupies a middle position between Ti and r 2 . This hope applies, of
course, only to the concept of value itself, but not to the reasonings which
lead to it; those were specific to Ti and r 2 , indeed different for I\ and for
r 2 , and altogether impracticable for T itself; i.e., we expect for the future
more from (14:A:d), (14:A:e), (14:B:d), (14:B:e) than from (14:A:a),
(14 :A :b), (14:B:a), (14:B:b).

1 Observe that the original game T was symmetric with respect to the two players
1 and 2, if we let each player take his function 3Ci(n, t 2 ), 3C 2 (ti, t 2 ) with him in an inter-
change; i.e. the personal moves of 1 and 2 had both the same character in T.

For a narrower concept of symmetry, where the functions 3Ci(n, r 2 ), 3C 2 (n, r 2 ) are
held fixed, cf. 14.6.

* This point deserves careful consideration: Naturally these two characterizations
must obtain from each other by interchanging the roles of the players 1 and 2. But in
this case the statements coincide also directly when no interchange of the players is made
at all. This is due to their individual symmetry.

STRICTLY DETERMINED GAMES

105

These are clearly only heuristic indications. Thus far we have not
even attempted the proof that a numerical value of a play can be defined
in this manner for T. We shall now begin the detailed discussion by which
this gap will be filled. It will be seen that at first definite and serious
difficulties seem to limit the applicability of this procedure, but that it will
be possible to remove them by the introduction of a new device (Cf. 14.7.1.
and 17.1.-17.3., respectively).

14.4. Conclusions

14 . 4 . 1 . We have seen that a perfectly plausible interpretation of the
value of a play determines this quantity as

Vi = Max tj Min Tj JC(ri, r 2 ),
v 2 = Min Tj Max Ti 3C (n, r 2 ),

for the games Ti, r 2 , respectively, as far as the player 1 is concerned. 1

Since the game Ti is less advantageous for 1 than the game r 2 — in Ti
he must make his move prior to, and in full view of, his adversary, while in
r 2 the situation is reversed — it is a reasonable conclusion that the value
of Ti is less than, or equal to (i.e. certainly not greater than) the value of r 2 .
One may argue whether this is a rigorous “proof.” The question whether
it is, is hard to decide, but at any rate a close analysis of the verbal argu-
ment involved shows that it is entirely parallel to the mathematical proof
of the same proposition which we already possess. Indeed, the proposition
in question,

Vi ^ v 2

coincides with (13 :A*) in 13.4.3. (The 0, x, y there correspond to our

3C, ti, r 2 .)

Instead of ascribing Vi, v 2 as values to two games T x and r 2 different
from T we may alternatively correlate them with T itself, under suitable
assumptions concerning the “intellect” of the players 1 and 2.

Indeed, the rules of the game T prescribe that each player must make
his choice (his personal move) in ignorance of the outcome of the choice
of his adversary. It is nevertheless conceivable that one of the players,
say 2, “finds out” his adversary; i.e., that he has somehow acquired the
knowledge as to what his adversary’s strategy is. 2 The basis for this
knowledge does not concern us; it may (but need not) be experience from
previous plays. At any rate we assume that the player 2 possesses this
knowledge. It is possible, of course, that in this situation 1 will change
his strategy; but again let us assume that, for any reason whatever, he
does not do it. 8 Under these assumptions we may then say that player 2
has “found out” his adversary.

1 For player 2 the values are consequently — Vi, — v a .

* In the game T — which is in the normalized form — the strategy is just the actual
choice at the unique personal move of the player. Remember how this normalized form
was derived from the original extensive form of the game; consequently it appears that
this choice corresponds equally to the strategy in the original game.

* For an interpretation of all these assumptions, cf. 17.3.1.

106

ZERO-SUM TWO-PERSON GAMES: THEORY

In this case, conditions in T become exactly the same as if the game were
Ti, and hence all discussions of 14.3.1. apply literally.

Similarly, we may visualize the opposite possibility, that player 1 has
“ found out” his adversary. Then conditions in T become exactly the same
as if the game were r 2 ; and hence all discussions of 14.3.3. apply literally.

In the light of the above we can say:

The value of a play of the game r is a well-defined quantity if one of the
following two extreme assumptions is made: Either that player 2 “finds
out” his adversary, or that player 1 “finds out” his adversary. In the
first case the value of a play is Vi for 1, and — Vi for 2; in the second case
the value of a play is v 2 for 1 and — v 2 for 2.

14 . 4 . 2 . This discussion shows that if the value of a play of T itself —
without any further qualifications or modifications — can be defined at all,
then it must lie between the values of Vi and v 2 . (We mean the values
for the player 1.) I.e. if we write v for the hoped-for value of a play of T
itself (for player 1), then there must be

Vi ^ v ^ v 2 .

The length of this interval, which is still available for v, is

A = v 2 — Vi ^ 0.

At the same time A expresses the advantage which is gained (in the
game T) by “finding out” one’s adversary instead of being “found out”
by him. 1

Now the game may be such that it does not matter which player “finds
out” his opponent; i.e., that the advantage involved is zero. According
to the above, this is the case if and only if

A = 0

or equivalently

V i = V 2

Or, if we replace Vi, v 2 by their definitions:

Max Ti Min T| 3C(ri, r 2 ) = Min T2 Max Ti X(ii, r 2 ).

If the game T possesses these properties, then we call it strictly determined.

The last form of this condition calls for comparison with (13:3) in 13.3.1.
and with the discussions of 13.4.1.-13.5.2. (The </>, x, y there again corre-
spond to our 3C, 7i, r 2 ). Indeed, the statement of (13 :B*) in 13.4.3. says
that the game T is strictly determined if and only if a saddle point of
3C(ri, t 2 ) exists.

14.5. Analysis of Strict Determinateness

14 . 6 . 1 . Let us assume the game r to be strictly determined; i.e. that a
saddle point of 3C(ri, r 2 ) exists.

1 Observe that this expression for the advantage in question applies for both players:
The advantage for the player 1 is v 2 — Vi; for the player 2 it is ( - Vi) — ( — v 2 ) and these
two expressions are equal to each other, i.e. to A.

STRICTLY DETERMINED GAMES

107

In this case it is to be hoped — considering the analysis of 14.4.2. — that
it will be possible to interpret the quantity

v = Vi = v 2

as the value of a play of r (for the player 1). Recalling the definitions of
Vi, v 2 and the definition of the saddle value in 13.4.3. and using (13:0*) in
13.5.2., we see that the above equation may also be written as

v = Max Ti Min Tj 3C(n, r 2 ) = Min Tj Max^ 0 C(ti, t 2 )

= Sa ri / Tj 5C(ri, t 2 ).

By retracing the steps made at the end of 14.3.1. and at the end of 14.3.3.
it is indeed not difficult to establish that the above can be interpreted as
the value of a play of T (for player 1).

• Specifically: (14:A:c), (14:A:d), (14:A:e), (14:B:c), (14:B:d), (14:B:e)
of 14.3.1. and 14.3.3. where they apply to lb and r 2 respectively, can now
be obtained for T itself. We restate first the equivalent of (14:A:d) and
(14:B:d):

(14:C:d) Player 1 can, by playing appropriately, secure for himself

a gain ^ v, irrespective of what player 2 does.

Player 2 can, by playing appropriately, secure for himself
a gain ^ — v irrespective of what player 1 does.

In order to prove this, we form again the set A of (14:A:a) in 14.3.1.
and the set B of (14:B:a) in 14.3.3. These are actually the sets A+, B+ of
13.5.1. (the <t> corresponds to our 3C). We repeat:

(14:D:a) A is the set of those ri for which Min rj 3C(ri, r 2 ) assumes its

maximum value; i.e. for which

Min Tj 3C(ri, r 2 ) = Max Tj Min Tj 3 C(ti, t 2 ) = v.

(14:D:b) B is the set of those r 2 for which Max Tj 3C(ri, r 2 ) assumes

its minimum value; i.e. for which

Max Ti 3C(ri, t 2 ) = Min Tj Max Ti 0 C(ti, r 2 ) = v.

Now the demonstration of (14:C:d) is easy:

Let player 1 choose ti from A. Then irrespective of what player 2

does, i.e. for every t 2 , we have 3C(ri, r 2 ) ^ Min T X(r i, r 2 ) = v, i.e., Psgain

« *

is ^ v.

Let player 2 choose r 2 from B. Then, irrespective of what player 1
does, i.e. for every r X) we have JC(ri, r 2 ) ^ Max Tj 3C(r h t 2 ) = v, i.e. l's gain
is ^ v and so 2^s gain is ^ — v.

This completes the proof.

We pass now to the equivalent of (14:A:e) and (14:B:e). Indeed,
(14:C:d) as formulated above can be equivalently formulated thus:
(14:C:e) Player 2 can, by playing appropriately, make it sure that

the gain of player 1 is ^ v, i.e. prevent him from gaining > v
irrespective of what player 1 does.

108

ZERO-SUM TWO-PERSON GAMES: THEORY

Player 1 can, by playing appropriately, make it sure thal
the gain of player 2 is ^ — v i.e. present him from gaining
> — v irrespective of what player 2 does.

(14:C:d) and (14:C:e) establish satisfactorily our interpretation of v as the
value of a play of T for the player 1, and of — v for the player 2.

14 . 5 . 2 . We consider now the equivalents of (14:A:a), (14:A:b), (14:B:a),
(14:B:b).

Owing to (14:C:d) in 14.5.1. it is reasonable to define a good way for 1
to play the game T as one which guarantees him a gain which is greater
than or equal to the value of a play for 1, irrespective of what 2 does; i.e. a
choice of n for which 3C(ri, t 2 ) ^ v for all r 2 . This may be equivalently
stated as Min Tj 3C(ri, t 2 ) ^ v.

Now we have always Min Tj 3C(ri, r 2 ) g Max Tj Min Tj 3 C(ti, t 2 ) = v.
Hence the above condition for ti amounts to Min Tj 3C(ri, r 2 ) = v, i.e.
(by (14:D:a) in 14.5.1.) to n being in A.

Again, by (14:C:d) in 14.5.1. it is reasonable to define the good way for
2 to play the game r as one which guarantees him a gain which is greater
than or equal to the value of a play for 2, irrespective of what 1 does; i.e.
a choice of r 2 for which — JC(r i, t 2 ) ^ — v for all n. That is, 5C(r i, r 2 ) ^ v
for all r i. This may be equivalently stated as Max Tj 3C(ri, t 2 ) ^ v.

Now we have always Max Tj 0 C(ti, t 2 ) ^ Min Tj Max T> 0 C(ti, r 2 ) = v.
Hence the above conditions for r 2 amounts to Max Ti 3 C(t!, r 2 ) = v, i.e. (by
(14:D:b) in 14.5.1.) to r 2 being in B.

So we have:

(14:C:a) The good way (strategy) for 1 to play the game T is to

choose any r i belonging to A , — A being the set of (14:D:a) in
14.5.1.

(14:C:b) The good way (strategy) for 2 to play the game r is to

choose any r 2 belonging to B, — B being the set of (14:D:b)
in 14.5.1. 1

Finally our definition of the good way of playing, as stated at the
beginning of this section, yields immediately the equivalent of (14:A:c)
or (14:B:c):

(14:C:c) If both players 1 and 2 play the game T well — i.e. if ri

belongs to A and r 2 belongs to B — then the value of 3C(ri, r 2 )
will be equal to the value of a play (for 1), i.e. to v.

We add the observation that (13 :D*) in 13.5.2. and the remark concerning
the sets A, B before (14:D:a), (14:D:b) in 14.5.1. together give this:

(14:C:f) Both players 1 and 2 play the game T well — i.e. ri belongs

to A and r 2 belongs to B — if and only if n, r 2 is a saddle point
of 3C(ri, t 2 ).

1 Since this is the game r each player must make his choice (of n or r 2 ) without
knowledge of the other player’s choice (of tj or n). Contrast this with (14:A:b) in
14.3.1. for Tx and with (14:B:b) in 14.3.3. for IY

STRICTLY DETERMINED GAMES

109

14.6. The Interchange of Players. Symmetry

14.6. (14:C:a)-(14:C:f) in 14.5.1. and 14.5.2. settle everything as far
as the strictly determined two-person games are concerned. In this
connection let us remark that in 14.3.1., 14.3.3. — for Ti, r 2 — we derived
(14:A:d), (14:A:e), (14:B:d), (14:B:e) from (14:A:a), (14:A:b), (14:B:a),
(14:B:b) while in 14.5.1., 14.5.2. — for T itself — we obtained (14:C:a),
(14:C:b) from (14:C:d), (14:C:e). This is an advantage since the argu-
ments of 14.3.1., 14.3.3. in favor of (14:A:a), (14:A:b), (14:B:a), (14:B:b)
were of a much more heuristic character than those of 14.5.1., 14.5.2. in
favor of (14:C:d), (14:C:e).

Our use of the function 3C(ri, t 2 ) = 3Ci(ri, r 2 ) implies a certain asymmetry
of the arrangement; the player 1 is thereby given a special role. It ought
to be intuitively clear, however, that equivalent results would be obtained
if we gave this special role to the player 2 instead. Since interchanging
the players 1 and 2 will play a certain role later, we shall nevertheless give a
brief mathematical discussion of this point also.

Interchanging the players 1 and 2 in the game T — of which we need
not assume now that it is strictly determined — amounts to replacing the
functions 3Ci(ri, r 2 ), 3C 2 (ti, t 2 ) by 3C 2 (t 2 , r 1 ), 3Ci(t 2 , n). 1,2 It follows, there-
fore, that this interchange means replacing the function 3C (r 1 , r 2 ) by — 3C (r 2 , r 1 ) .

Now the change of sign has the effect of interchanging the operations
Max and Min. Consequently the quantities

Max Tj Min Tj 0C(r,, 7 2 ) = Vi,

Min T| Max Tj 3 C(ti, t 2 ) = v 2 ,

as defined in 14.4.1. become now

Max T| Min Tj [— 3 C(t 2 , ti)] = — Min Ti Max Tj 3 C(t 2 , ri)

= — Min Tj Max Tj JC(ri, r 2 ) 3 = ~v 2 .
Min r# Max Tj [ — JC(r 2 , ti)] = — Max Tj Min Ti 0C(r 2 , n)

= — Max Ti Min Tj 0C(ri, r 2 ) 3 = — Vi.

So Vi, v 2 become — v 2 , — Vi. 4 Hence the value of

A = v 2 - vj = (-Vi) - ( v 2 )

1 This is no longer the operation of interchanging the players used in 14.3.4. There
we were only interested in the arrangement and the state of information at each move,
and the players 1 and 2 were considered as taking their functions 3Ci(n, r 2 ) and 3C*(n, n)
with them (cf. footnote 1 on p. 104). In this sense r was symmetric, i.e. unaffected by
that interchange (id.).

At present we interchange the roles of the players 1 and 2 completely, even in their
functions Xi(n, t 2 ) and3C 2 (n, r 2 ).

1 We had to interchange the variables n, r 2 since n represents the choice of player 1
and r 2 the choice of player 2. Consequently it is now r 2 which has the domain 1, • • • , 0i.
Thus it is again true for 3 C*(t 2 , n) — as it was before for3C*(n, t 2 ) — that the variable before
the comma has the domain 1, • • • , 0i and the variable after the comma, the domain
1, • • * , 0*.

8 This is a mere change of notations: The variables n, r 2 are changed around to r 2 , n.

4 This is in harmony with footnote 1 on p. 105, as it should be.

110 ZERO-SUM TWO-PERSON GAMES: THEORY

is unaffected, 1 and if T is strictly determined, it remains so, since this prop-
erty is equivalent to A = 0. In this case v = Vi = v 2 becomes

— V = — Vi = — V2-

It is now easy to verify that all statements (14:C:a)-(14:C:f) in 14.5.1.,
14.5.2. remain the same when the players 1 and 2 are interchanged.

14.7. Non-strictly Determined Games

14 . 7 . 1 . All this disposes completely of the strictly determined games,
but of no others. For a game I which is not strictly determined we have
A > 0 i.e. in such a game it involves a positive advantage to “find out”
one's adversary. Hence there is an essential difference between the
results, i.e. the values in Ti and in r 2 , and therefore also between the good
ways of playing these games. The considerations of 14.3.1., 14.3.3. provide
therefore no guidance for the treatment of T. Those of 14.5.1., 14.5.2.
do not apply either, since they make use of the existence of saddle points
of 3C(ri, t 2 ) and of the validity of

Max Ti Min Tj i, r 2 ) = Min Tj Max Ti 3C(ti, t 2 ),

i.e. of T being strictly determined. There is, of course, some plausibility
in the inequality at the beginning of 14.4.2. According to this, the value v
of a play of T (for the player 1) — if such a concept can be formed at all in
this generality, for which we have no evidence as yet 2 — is restricted by

Vi ^ v ^ v 2 .

But this still leaves an interval of length A = v 2 — Vi > 0 open to v;
and, besides, the entire situation is conceptually most unsatisfactory.

One might be inclined to give up altogether: Since there is a positive
advantage in “finding out” one's opponent in such a game T, it seems
plausible to say that there is no chance to find a solution unless one makes
some definite assumption as to “who finds out whom,” and to what extent. 3

We shall see in 17. that this is not so, and that in spite of A > 0 a solu-
tion can be found along the same lines as before. But we propose first,
without attacking that difficulty, to enumerate certain games T with A > 0,
and others with A = 0. The first — which are not strictly determined —
will be dealt with briefly now; their detailed investigation will be under-
taken in 17.1. The second — which are strictly determined — will be ana-
lyzed in considerable detail.

14 . 7 . 2 . Since there exist functions JC(ri, r 2 ) without saddle points (cf.
13.4.1., 13.4.2.; the <£(x, y) there, is our 3C(ri, t 2 )) there exist not strictly
determined games T. It is worth while to re-examine those examples — i.e.

1 This is in harmony with footnote 1 on p. 106, as it should be.

2 Cf. however, 17.8.1.

2 In plainer language: A > 0 means that it is not possible in this game for each player
simultaneously to be cleverer than his opponent. Consequently it seems desirable to
know just how clever each particular player is.

STRICTLY DETERMINED GAMES

111

the functions described by the matrices of Figs. 12, 13 on p. 94 — in the
light of our present application. That is, to describe explicitly the games
to which they belong. (In each case, replace </>(z, y) by our JC(ri, r 2 ), r 2
being the column number and n the row number in every matrix. Cf. also
Fig. 15 on p. 99).

Fig. 12: This is the game of “ Matching Pennies.” Let — for ri and for
r 2 — 1 be “ heads” and 2 be “ tails, ” then the matrix element has the value 1
if n, r 2 “match” — i.e. are equal to each other — and —1, if they do not.
So player 1 “matches” player 2: He wins (one unit) if they “match” and
he loses (one unit), if they do not.

Fig. 13: This is the game of “Stone, Paper, Scissors.” Let — for ri and for
r 2 — 1 be “stone,” 2 be “paper,” and 3 be “scissors.” The distribution of
elements 1 and — 1 over the matrix expresses that “paper” defeats “stone,”
“scissors” defeat “paper,” “stone” defeats “scissors.” 1 Thus player 1
wins (one unit) if he defeats player 2, and he loses (one unit) if he is defeated.
Otherwise (if both players make the same choice) the game is tied.

14 . 7 . 3 . These two examples show the difficulties which we encounter
in a not strictly determined game, in a particularly clear form; just because
of their extreme simplicity the difficulty is perfectly isolated here, in vitro.
The point is that in “Matching Pennies” and in “Stone, Paper, Scissors,”
any way of playing — i.e. any n or any r 2 — is just as good as any other:
There is no intrinsic advantage or disadvantage in “heads” or in “tails”
per se , nor in “stone,” “paper” or “scissors” per se. The only thing which
matters is to guess correctly what the adversary is going to do ; but how are
we going to describe that without further assumptions about the played
“intellects”? 2

There are, of course, more complicated games which are not strictly
determined and which are important from various more subtle, technical
viewpoints (cf. 18., 19.). But as far as the main difficulty is concerned,
the simple games of “Matching Pennies” and of “Stone, Paper, Scissors”
are perfectly characteristic.

14.8. Program of a Detailed Analysis of Strict Determinateness

14 . 8 . While the strictly determined games T — for which our solution
is valid — are thus a special case only, one should not underestimate the size
of the territory they cover. The fact that we are using the normalized
form for the game T may tempt to such an underestimation: It makes things
look more elementary than they really are. One must remember that the
r i, r 2 represent strategies in the extensive form of the game, which may be
of a very complicated structure, as mentioned in 14.1.1.

In order to understand the meaning of strict determinateness, it is
therefore necessary to investigate it in relation to the extensive form of the
game. This brings up questions concerning the detailed nature of the moves,

1 “ Paper covers the stone, scissors cut the paper, stone grinds the scissors.”

2 As mentioned before, we shall show in 17.1. that it can be done.

112

ZERO-SUM TWO-PERSON GAMES: THEORY

— chance or personal — the state of information of the players, etc. ; i.e. we
come to the structural analysis based on the extensive form, as mentioned
in 12.1.1.

We are particularly interested in those games in which each player who
makes a personal move is perfectly informed about the outcome of the
choices of all anterior moves. These games were already mentioned in

6.4.1. and it was stated there that they are generally considered to be of a
particular rational character. We shall now establish this in a precise
sense, by proving that all such games are strictly determined. And this
will be true not only when all moves are personal, but also when chance
moves too are present.

16. Games with Perfect Information
15.1. Statement of Purpose. Induction

16.1.1. We wish to investigate the zero-sum two-person games somewhat
further, with the purpose of finding as wide a subclass among them as
possible in which only strictly determined games occur; i.e. where the
quantities

v i = Max Ti Min Tj 3C (n, r 2 ),
v 2 = Min Tj Max Ti 3C(ri, t 2 )

of 14.4.1. — which turned out to be so important for the appraisal of the
game — fulfill

Vi = v 2 = v.

We shall show that when perfect information prevails in T — i.e. when
preliminarity is equivalent to anteriority (cf. 6.4.1. and the end of 14.8.) —
then T is strictly determined. We shall also discuss the conceptual sig-
nificance of this result (cf. 15.8.). Indeed, we shall obtain this as a special
case of a more general rule concerning Vi, v 2 , (cf. 15.5.3.).

We begin our discussions in even greater generality, by considering a
perfectly unrestricted general n-person game T. The greater generality
will be useful in a subsequent instance.

16.1.2. Let T be a general n- person game, given in its extensive form.
We shall consider certain aspects of T, first in our original pre-set-theoretical
terminology of 6., 7., (cf. 15.1.), and then translate everything into the par-
tition and set terminology of 9., 10. (cf. 15.2., et sequ.). The reader will
probably obtain a full understanding with the help of the first discussion
alone; and the second, with its rather formalistic machinery, is only under-
taken for the sake of absolute rigor, in order to show that we are really
proceeding strictly on the basis of our axioms of 10.1.1.

We consider the sequence of all moves in T: 9Ri, 9H 2 , • • * , 9n„. Let
us fix our attention on the first move, 9Tli, and the situation which exists
at the moment of this move.

Since nothing is anterior to this move, nothing is preliminary to it
either; i.e. the characteristics of this move depend on nothing, — they are
constants. This applies in the first place to the fact, whether SHli is a chance

GAMES WITH PERFECT INFORMATION

113

move or a personal move ; and in the latter case, to which player 9dli belongs, —
i.e. to the value of k\ = 0, 1, • • • , n respectively, in the sense of 6.2.1.
And it applies also to the number of alternatives a x at 9Tli and for a chance
move (i.e. when k x = 0) to the values of the probabilities pi(l), • • • , pi(«i).
The result of the choice at 9TZi — chance or personal — is a a x = 1, • • • , on.

Now a plausible step suggests itself for the mathematical analysis
of the game T, which is entirely in the spirit of the method of “ complete
induction” widely used in all branches of mathematics. It replaces, if
successful, the analysis of T by the analysis of other games which contain
one move less than T. 1 This step consists in choosing a 9\ = 1, • • • , a\
and denoting by V 9i a game which agrees with T in every detail except that
the move 9Tli is omitted, and instead the choice <n is dictated (by the rules
of the new game) the value a x = 9 X 2 V 9x has, indeed, one move less than
T : Its moves are 9112, * * * , 9Tl„. 3 And our “inductive” step will have been
successful if we can derive the essential characteristics of T from those of
all I\, fri = 1, • • • , ot x .

15.1.3. It must be noted, however, that the possibilities of forming
r 9l are dependent upon a certain restriction on r. Indeed, every player
who makes a certain personal move in the game V 9i must be fully informed
about the rules of this game. Now this knowledge consists of the knowledge
of the rules of the original game r plus the value of the dictated choice at
9Tli, i.e. 9 X . Hence T 9y can be formed out of T — without modifying the rules
which govern the player's state of information in r — only if the outcome
of the choice at 9Tli, by virtue of the original rules T, is known to every
player at any personal move of his 3TC 2 , • • • , 9T l p ; i.e. 9Tli must be prelim-
inary to all personal moves 9Tl 2 , • • • , 9TI,. We restate this:

(15 :A) T 9i can be formed — without essentially modifying the

structure of r for that purpose — only if T possess the following
property:

(15:A:a) 2THi is preliminary to all personal moves 9112, * * * , 9TC*. 4

1 I.e. have v — 1 instead of v. Repeated application of this “inductive" step — if
feasible at all — will reduce the game r to one with 0 steps; i.e. to one of fixed, unalterable
outcome. And this means, of course, a complete solution for r. (Cf. (15:C:a) in 15.6.1.)

* E.g. T is the game of Chess, and <?i a particular opening move — i.e. choice at 9Hi — of
“white," i.e. player 1. Then is again Chess, but beginning with a move of the char-
acter of the second move in ordinary Chess — a “black," player 2 — and in the position
created by the “opening move" d\. This dictated “opening move" may, but need not,
be a conventional one (like E2-E4).

The same operation is exemplified by forms of Tournament Bridge where the
“umpires" assign the players definite — known and previously selected — “hands."
(This is done, e.g., in Duplicate Bridge.)

In the first example, the dictated move 9Tli was originally personal (of “white,"
player 1); in the second example it was originally chance (the “deal").

In some games occasionally “handicaps" are used which amount to one or more
such operations.

* We should really use the indices 1, •••,* — 1 and indicate the dependence on Ji;
e.g. by writing 3TCJ 1 , • • • , 9H^ r But we prefer the simpler notation 9R*, • • • , 3Tlr.

4 This is the terminology of 6.3. ; i.e. we use the special form of dependence in the
sense of 7.2.1. Using the general description of 7.2.1. we must state (15:A:a) like this:
For every personal move 3Tl«, «c — 2, • • • , v, the set contains the function <n.

114

ZERO-SUM TWO-PERSON GAMES: THEORY

16.2. The Exact Condition (First Step)

15.2.1. We now translate 15.1.2., 15.1.3. into the partition and set
terminology of 9., 10., (cf. also the beginning of 15.1.2.). The notations of
10.1. will therefore be used.

Oti consists of the one set 12 ((10:1 :f) in 10.1.1.), and it is a subpartition
of (&i ((10:1 :a) in 10.1.1.); hence (Bi too consists of the one set S2 (the others
being empty). 1,2 That is:

d /*.\ _ ( ft for precisely one k, say k = k h

A ) ( © for all k * ki.

This fci = 0, 1, • • • , n determines the character of Sflli; it is the k i of 6.2.1.
If k\ = 1, • • • , n — i.e. if the move is personal — then is also a subpar-
tition of 3D i(fci), ((10:1 :d) in 10.1.1. This was only postulated within Bi(k x ) }
but Bi(ki) = 12). Hence $>i(fti) too consists of the one set 12. 1 * 3 And for
k t* k h the £>i (k) which is a partition in B x (k) = © ((10:A:g) in 10.1.1.)
must be empty.

So we have precisely one A i of Oti, which is 12, and for k\ = 1, • • • , n
precisely one D i in all SDi(/c), which is also 12; while for ki = 0 there are no
Di in all £)](&).

The move 3Tli consists of the choice of a C i from Gi(k } )\ by chance if
fci = 0; by the player fci if fci = 1, • • • , n. C i is automatically a subset
of the unique A i(= 12) in the former case, and of the unique D j(= 12)
in the latter. The number of these C\ is au (cf. 9. 1 .5., particularly footnote 2
on p. 70); and since the A i or D x in question is fixed, this a x is a definite
constant, ai is the number of alternatives at 9fTli, the a x of 6.2.1. and 15.1.2.

These C i correspond to the — 1, * • • , a x of 15.1.2., and we denote
them accordingly by Ci(l), • • • ,Ci(ai). 4 Now (10:1 :h) in 10.1.1. shows —
as is readily verified — that (t 2 is also the set of the C i(l), • • • , Ci(ai), i.e.
equal to Ci.

So far our analysis has been perfectly general, — valid for c JIZi (and to a
certain extent for ffl 2 ) of any game 1\ The reader should translate these
properties into everyday terminology in the sense of 8.4.2. and 10.4.2.

We pass now to iy. This should obtain from r by dictating the move
Sflli — as described in 15.1.2. — by putting <n = Q\. At the same time the
moves of the game are restricted to 2TC 2 , * • , 31Z„. This means that the

1 This (Bi is an exception from (8:B:a) in 8.3.1.; cf. the remark concerning this (8:B:a)
in footnote 1 on p. 63, and also footnote 4 on p. 69.

8 Proof: Q belongs to Cti, which is a subpartition of <B X ; hence 0 is a subset of an
element of (Bi. This element is necessarily equal to O. All other elements of (Bi are
therefore disjunct from Q (cf. 8.3.1.), i.e. empty.

1 Cti, SDi(fci) unlike ffii (cf. above) must fulfill both (8:B:a), (8:B:b) in 8.3.1.; hence
both have no further elements besides U.

4 They represent the alternatives fti( 1), • • • , adai) of 6.2. and 9.1.4., 9.1.5.

GAMES WITH PERFECT INFORMATION

115

element ir — which represents the actual play — can no longer vary over all
12, but is restricted to C i(fri). And the partitions enumerated in 9.2.1. are
restricted to those with k = 2, • • • , v, 1 (and k = v + 1 for Ct,).

15.2.2. We now come to the equivalent of the restriction of 15.1.3.

The possibility of carrying out the changes formulated at the end of
15.2.1. is dependent upon a certain restriction on T.

As indicated, we wish to restrict the play — i.e. ir — within C i(fri). There-
fore all those sets which figured in the description of T and which were
subsets of 12, must be made over into subsets of Ci(fri) — and the partitions
into partitions within C i(fri) (or within subsets of C i(frj)). How is this to
be done?

The partitions which make up the descriptions of T (cf. 9.2.1.) fall into
two classes: those which represent objective facts — the Ct„ the (B« = (5,( 0),
5,( 1), * * * , 5,(n)) and the C ,(&)> k = 0, 1, • • • , n — and those which
represent only the player’s state of information, 2 the 2D,(fc), k = 1, * • • , n.
We assume, of course k ^ 2 (cf. the end of 15.2.1.).

In the first class of partitions we need only replace each element by
its intersection with C Thus (B, is modified by replacing its ele-
ments 5,(0), 5,(1), • • • , 5,(n) by Cj(*i) n 5,(0), C i(*i) n 5,(1), • • • ,
C i(fri) n 5,(n). In a, even this is not necessary: It is a subpartition of
Gfc 2 (since k ^ 2, cf. 10.4.1.), i.e. of the system of pairwise disjunct sets
(C i(l), • • • , Ci(a{)) (cf. 15.2.1.); hence we keep only those elements of a,
which are subsets of Ci(fri), i.e. that part of Gt, which lies in C i(fri). The
<3,(fc) should be treated like (B, but we prefer to postpone this discussion.

In the second class of partitions — i.e. for the D,(fc) — we cannot do any-
thing like it. Replacing the elements of 3D,(/c) by their intersections with
C i(fri) would involve a modification of a player’s state of information 3 and
should therefore be avoided. The only permissible procedure would be
that which was feasible in the case of ft,: replacement — of 3 X(k ) — by that
part of itself which lies in C i(fri). But this is applicable only if 3D ,(fc) — like
a, before — is a subpartition of Ct 2 (for k ^ 2). So we must postulate this.

Now e.(fc) takes care of itself: It is a subpartition of 3D,(fc) ((10:1 :c)
in 10.1.1.), hence of Ct 2 (by the above assumption); and so we can replace
it by that part of itself which lies in Ci(fri).

So we see: The necessary restriction of T is that every 3D,(fc) (with
k S 2) must be a subpartition of (t 2 . Recall now the interpretation of
8.4.2. and of (10:A:d*), (10:A:g*) in 10.1.2. They give to this restriction
the meaning that every player at a personal move 9n 2 , • * • , is fully

1 We do not wish to change the enumeration to k - 1, • • • , v — 1, cf. footnote 3
on p. 113.

2 a, represents the umpire’s state of information, but this is an objective fact: the
events up to that moment have determined the course of the play precisely to that
extent (cf. 9.1.2.).

3 Namely, giving him additional information.

116

ZERO-SUM TWO-PERSON GAMES: THEORY

informed about the state of things after the move 9Tli (i.e. before the move
911*) expressed by 0t 2 . (Cf. also the discussion before (10:B) in 10.4.2.)
That is, 9Tli must be preliminary to all moves 9Tl 2 , * * • , 9TI,.

Thus we have again obtained the condition (15:A:a) of 15.1.3. We
leave to the reader the simple verification that the game iy fulfills the
requirements of 10.1.1.

16.3. The Exact Condition (Entire Induction)

16 . 3 . 1 . As indicated at the end of 15.1.2., we wish to obtain the char-
acteristics of T from those of all iy, 9i = 1, • • * , ai, since this — if suc-
cessful — would be a typical step of a “ complete induction.”

For the moment, however, the only class of games for which we possess
any kind of (mathematical) characteristics consists of the zero-sum two-
person games: for these we have the quantities Vi, v 2 (cf. 15.1.1.). Let us
therefore assume that T is a zero-sum two-person game.

Now we shall see that the Vi, v 2 of T can indeed be expressed with the
help of those of the V 9i , 9i = 1, • • • , a\ (cf. 15.1.2.). This circumstance
makes it desirable to push the “induction” further, to its conclusion:

i.e., to form in the same way r*^, • • • , The

point is that the number of steps in these games decreases successively

from v (for T), v — 1 (for T 9i ), over v — 2, v — 3, • • • to 0 (for r#^ »,);

i.e. r* if * f 9 V is a “vacuous” game (like the one mentioned in the

footnote 2 on p. 76). There are no moves; the player k gets the fixed
amount • • • , 9,).

This is the terminology of 15.1.2., 15.1.3., — i.e. of 6., 7. In that of
15.2.1., 15.2.2. — i.e. of 9., 10. — we would say that ft (for T) is gradually
restricted to a Ci(fri) of a 2 (for T 9i ), a C 2 (di, d 2 ) of a 3 (for r* it * f ), a Cz(9 h 9 2f 9 2 )
of 0>4 (for r fiiff(fj ), etc., etc., and finally to a C,(9 h d 2 , • • • , 9 ,) of Ct„+i (for

r fi .f t f r ). And this last set has a unique element ((10:1 :g) in 10.1.1.),

say fr. Hence the outcome of the game 9p is fixed: The player &

gets the fixed amount $*(*).

Consequently, the nature of the game r» » f , is — trivially — clear;

it is clear what this game’s value is for every player. Therefore the process
which leads from the T 9i to T — if established — can be used to work back-
wards from r fliff 9 , to w ¥ x to r, iifi t etc., etc., to

to V 9i and finally to T.

But this is feasible only if we are able to form all games of the sequence

r#i, I\,* a , • • • , I\,, t 9v , i.e. if the final condition of 15.1.3.

or 15.2.2. is fulfilled for all these games. This requirement may again be
formulated for any general n- person game T; so we return now to those T.

16 . 3 . 2 . The requirement then is, in the terminology of 15.1.2., 15.1.3.
(i.e. of 6., 7.) that 9Ei must be preliminary to all 9fR 2 , 9TC 3 , • • • , that

1 ** 1, • • • , a\) 5% * 1, • * * , «2 where a* ** a*(£i); *= 1, • • • , a* where

<*i ** 9%)\ etc., etc.

GAMES WITH PERFECT INFORMATION

117

STl* must be preliminary to all 9TC 3 , 91t 4 , • * • , 9TC,; etc., etc.; i.e. that pre-
liminarity must coincide with anteriority.

In the terminology of 15.2.1., 15.2.2. — i.e. of 9., 10. — of course the same
is obtained: All 2D«(fc), * ^ 2 must be subpartitions of 0fc 2 ; all 2D*(fc), k ^ 3
must be subpartitions of 0t 3 , etc., etc.; i.e. all 3D K (k) must be subpartitions
of 0\ if k X. 1 Since 0 K is a subpartition of 0\ in any case (cf. 10.4.1.), it
suffices to require that all 3D<(fc) be subpartitions of O k . However 0 K is a
subpartition of 3D*(/c) within (B K (k) ((10:1 :d) in 10.1.1.); consequently our
requirement is equivalent to saying that 3D K (k) is that part of 0 K which lies in
B«(fc). 2 By (10:B) in 10.4.2. this means precisely that preliminarity and
anteriority coincide in T.

By all these considerations we have established this:

(15:B)

(15:1)

of

In order to be able to form the entire sequence of games

r, f$ , r* 9 , r* 9 9 , * * , r# 9 >

) V*** 1 9 i ,9 f * * * ,9 p

V, V - 1, V - 2,

,0

moves respectively, it is necessary and sufficient that in the game
T preliminarity and anteriority should coincide, — i.e. that per-
fect information should prevail. (Cf. 6.4.1. and the end of 14.8.)

If F is a zero-sum two-person game, then this permits the
elucidation of T by going through the sequence (15:1) back-
wards — from the trivial game F» it » |t . . . , 9p to the significant
game T — performing each step with the help of the device which
leads from the T 9l to T as will be shown in 15.6.2.

16.4. Exact Discussion of the Inductive Step

15.4.1. We now proceed to carry out the announced step from the r #i *
to T, the “ inductive step.” T need therefore fullfill only the final condition
of 15.1.3. or 15.2.2., but it must be a zero-sum two-person game.

Hence we can form all <t i = 1, • • • , a i, and they also are zero-sum two-
person games. We denote the two played strategies in T by Sj, • • • , 2? 1
and 2' 2 , • • • , 2?*; and the “mathematical expectation” of the outcome of
the play for the two players, if the strategies 2^, 2 2 * are used, by

3Ci(ti, t 2 ) = 3C(ri, t 2 ), 3C 2 (tj, t 2 ) as — 3 C(ti, t 2 )

(cf. 11.2.3. and 14.1.1.). We denote the corresponding quantities in T #i ,
by z; t/1> • • • , 2%? and Z^ /2> ■ ■ ■ , 2*';, / \ and if the strategies 2^',
are used, by

r » 1 /j) = r.,/}), 3C», (t.j/ 1 , T #i /j).

1 We stated this above for X = 2, 3, • • • ; for X = 1 it is automatically true: every
partition is a subpartition of Ott since Oti consists of the one set ft ((10:1 :f) in 10.1.1.).

* For the motivation — if one is wanted — cf. the argument of footnote 3 on p. 63.

* From now on we write <n, v*, • • • , instead of 9i, £*,••',£» because no mis-
understandings will be possible.

118

ZERO-SUM TWO-PERSON GAMES: THEORY

We form the Vi, v» of 14.4.1. for T and for r #i denoting them in the latter
case by v #i /i, v, i n . So

v i = Max, ( Min ri 3C(n, t 2 ),
v* = Min,, Max Ti 3C(ri, t 2 ),

and

v»,/i = Max r<r(/i Min V/t 3C,/r, i/l( r, i/2 ),
v#,/t - Min r<r /| Max r , /i ac, i (r <ri/1 , t v »).

Our aim is to express the Vi, v 2 in terms of the v, x n, v, ( / 2 .

The ki of 15.1.2., 15.2.1. which determine the character of the move 9TCi
will play an essential role. Since n — 2, its possible values are A: i = 0, 1, 2.
We must consider these three alternatives separately.

15.4.2. Consider first the case k x = 0; i.e. let 31Xi be a chance move. The
probabilities of its alternatives ff! = 1, • • • , are the pi(l), • • • ,p x (a x )
mentioned in 15.1.2. (pi(<ri)isthepi(Ci)of (10:A:h) in 10.1.1. withCi = C x (<r x )
in 15.2.1.).

Now a strategy of player 1 in T, consists obviously in specifying a
strategy of player 1 in r v Z 9i x /X for every value of the chance variable

cri = 1, • • • ,<*1,4.6., the 2I» correspond to the aggregates 2%, • • • ,
for all possible combinations ry i, * * * , r tti / i.

Similarly a strategy of player 2 in T, 25* consists in specifying a strategy

of player 2 in r,„ ztf for every value of the chance variable a i = 1, • • • ,

c*i; i.e. the 2J* correspond to the aggregates 2%, * * * , 2 a “ l ^ for all possi-
ble combinations ri /2 , * • • , r aj/2 .

Now the “mathematical expectations” of the outcomes in I 1 and in l' 9i
are connected by the obvious formula

3C(ti, t 2 ) = PifrO^Ovi, *v i/2 ).

Therefore our formula for Vi gives
Vi = Max Tj Min Tj 3C(ri, t 2 )

= Max Vi X/1 Min v , „ £

* 1-1

«1

The (ri-term of the sum ^ on the extreme right-hand side

P\(<n)W* x ( r 'x n > r v 2 )

1 This is clear intuitively. The reader may verify it from the formalistic point of
view by applying the definitions of 11.1.1. and (11 : A) in 11.1.3. to the situation described
in 15.2.1.

GAMES WITH PERFECT INFORMATION

119

contains only the two variables r #t /i, t„ / 2 . Thus the variable pairs

Tl/l, T V 2] • • • ; Ta^/1, T aj /2

occur separately, in the separate crt-terms

= 1, * * * ; <ri = «i.

Hence in forming the Min Vj r<x we can minimize each <ri-term sep-
arately, and in forming the Max Vi Ta /x we can again maximize each

<ri-term separately. Accordingly, our expression becomes

£ pi(<ri) Max V /j T #/ *) = £ p t (<ri)v, i/t .

Thus we have shown

<*i

(15:2) vi= £ pi(ori)v ri /i.

<n-i

If the positions of Max and Min are interchanged, then literally the
same argument yields

(15:3) v 2 = £

15 . 4 . 3 . The case of fci = 1 comes next, and in this case we shall have
to make use of the result of 13.5.3. Considering the highly formal character
of this result, it seems desirable to bring it somewhat nearer to the reader's
imagination by showing that it is the formal statement of an intuitively
plausible fact concerning games. This will also make it clearer why this
result must play a role at this particular juncture.

The interpretation which we are now going to give to the result of 13.5.3.
is based on our considerations of 14.2.-14.5. — particularly those of 14.5.1.,
14.5.2. — and for this reason we could not propose it in 13.5.3.

For this purpose we shall consider a zero-sum two-person game T in its
normalized form (cf. 14.1.1.) and also its minorant and majorant games IT,
r 2 (cf. 14.2.).

If we decided to treat the normalized form of r as if it were an extensive
form, and introduced strategies etc. with the aim of reaching a (new)
normalized form by the procedure of 11.2.2., 11.2.3. then nothing would
happen, as described in 11.3. and particularly in footnote 1 on p. 84. The
situation is different, however, for the majorant and minorant games IT, IT;
these are not given in the normalized form, as mentioned in footnotes 2 and 3
on p. 100. Consequently it is appropriate and necessary to bring them into
their normalized forms — which we are yet to find — by the procedure of
11.2.2., 11.2.3.

120

ZERO-SUM TWO-PERSON GAMES: THEORY

Since complete solutions of T h r 2 were found in 14.3.1., 14.3.3., it is to
be expected that they will turn out to be strictly determined. 1

It suffices to consider Ti (cf. the beginning of 14.3.4.), and this we now
proceed to do.

We use the notations r h r 2 , 3C(n, r 2 ) and Vi, v 2 for T and we denote the
corresponding concepts for by r(, r' 2 , 3C '(rj, t 2 ) and \[> v 2 .

A strategy of player 1 in Ti consists in specifying a (fixed) value
n(= 1, • • • , j8i) while a strategy of player 2 in T i consists in specifying a
value of r 2 (= 1 , • • • , 0 2 ) depending on ri for every value of ri(= 1 , • • • ,
Pi). 2 So it is a function of ri : r 2 = 3 2 (ti).

Thus r\ is ti, while r 2 corresponds to the functions 3 2 , and X'(rJ, r' 2 ) to
3C(r,, 3 2 (ri)). Accordingly

vi = Max Ti Min 3i 3C(n, 3 2 (n)),
v 2 = Min 3f Max Ti 3C(ri, 3 2 (ri)).

Hence the assertion that Ti is strictly determined, i.e. the validity of
v[ = v' 2 coincides precisely with (13 :E) in 13.5.3. ; there we need only replace
the x, u,/(x), i(x y f(x )) by our n, r 2 , 3 2 (t,), 3C(ti, 3 2 (ti)).

This equivalence of the result of 13.5.3. to the strictly determined
character of Ti makes intelligible why 13.5.3. will play an essential role
in the discussion which follows. Ti is a very simple example of a game in
which perfect information prevails, — and these are the games which are the
ultimate goal of our present discussions (cf. the end of 15.3.2.). And the
first move in Ti is precisely of the kind which is coming up for discussion
now: It is a personal move of player 1, — i.e. ki = 1.

15.5. Exact Discussion of the Inductive Step (Continuation)

15 . 5 . 1 . Consider now the case ki = 1; i.e. let 3Tli be a personal move
of the player 1.

A strategy of player 1 in T, consists obviously in specifying a (fixed)
value <rj(= 1, • • • , c*i) and a (fixed) strategy of player 1 in I\®, *; i.e.

the 2]> correspond to the pairs <r$, r a \/ 1 .

1 This is merely a heuristic argument, since the principles on which the “solutions”
of 14.3.1., 14.3.3. are based are not entirely the same as those by which we disposed of the
strictly determined case in 14.5.1., 14.5.2., although the former principles were a stepping
stone to the latter. It is true that the argument could be made pretty convincing by an
“unmathematical,” purely verbal, amplification. We prefer to settle the matter
mathematically, the reasons being the same as given in a similar situation in 14.3.2.

* This is clear intuitively. The reader may verify it from the formalistic point of
view, by reformulating the definition of Ti as given in 14.2. in the partition and set
terminology, and then applying the definitions of 11.1.1. and (11 :A) in 11.1.3.

The essential fact is, at any rate, that in Ti the personal move of player 1 is pre-
liminary to that of player 2.

1 Cf. footnote 1 on p. 118 or footnote 2 above.

GAMES WITH PERFECT INFORMATION

121

A strategy of player 2 in F, on the other hand, consists in specifying
a strategy of player 2 in IV, X r <,‘ /t , for every value of the variable <r° =
1, • • • , a 1 . 1 So t,; / 2 isafunction of <rJ:r,;/ 2 = 3 2 (<rJ);i.e. the 25 * correspond
to the functions 3 2 and clearly

•3C(ti, t 2 ) = 3C,J(t»J/i, 3 2 (ff})).

Therefore our formula for Vi gives:

v i = Max,;, r<rJ/i Min 3] X,‘(r,* /h 3 2 (<r?))

= Max r<r , /1 Max,; Min 3| 3C,;(r,;„, 3 2 (<rJ)).

Now

Max,; Min 3i 3C,;(r,;/i, 3 2 (<t?)) = Max,; Min^JC ,;(r,;/,, t,; /2 )

owing to (13:G) in 13.5.3.; there we need only replace the x, u,f(x), 4/(x, u )
by our <r° u r,; /2 , 3 2 (<rJ), 3C,;(r,;/i, r,; /2 ). s Consequently

v, = Max,, ?/i Max,; Min, t,; /2 )

= Max,; Max v?/1 Min, i;/l 3C,;(r,;/,, r,; /2 )

= Max,; v,;,!

And our formula for v 2 gives : 3

v 2 = Min 3i Max.j.^o/.JC,;^,;/!, 3 2 (<r?))

= Min 3i Max,; Max,^^ 3C,;(r,;/i, 3 2 (<rJ)).

Now

Min 3j Max,; Max r<r;/i 3C,;(T,;,i, 3 2 (aJ))

= Max,; Min 3j Max f<r;/1 3C,;(r,;/i, 3 2 (<r$))

= Max,; Min v;/J Max T<r;/1 3C,;(T,;/i, r,; /2 )

owing to (13 :E) and (13 :G) in 13.5.3.; there we need only replace the x, u,
fix'), iix, u) by our <rj, r,; /2 , 3 2 (<rJ), Max^OC,;^,;/!, t,; / 2 ). 4 Consequently

v 2 = Max,; Min v;/1 Max^^X^r,;/,, r,; /2 )

= Max,; v,; /2 .

Summing up (and writing <n instead of tr°) :

1 Cf. footnote 1 on p. 118 or footnote 2 on p. 120.
t r<rJ/i must be treated in this case as a constant.

This step is of course a rather trivial one, — cf. the argument loc. cit.

3 In contrast to 15.4.2., there is now an essential difference between the treatments of
vi and Vi.

4 r//i is killed in this case by the operation Max* 0 /t .

1 r

This step is not trivial. It makes use of (13:E) in 13.5.3., i.e. of the essential result
of that paragraph, as stated in 15.4.3.

122

ZERO-SUM TWO-PERSON GAMES: THEORY

(15:4) Vi = Max ( , i v^/i,

(15:5) v 2 = Max^ v„ i/2 .

15 . 5 . 2 . Consider finally the case k i = 2; i.e. let 3Tli be a personal move of
player 2.

Interchanging players 1 and 2 carries this into the preceding case {k i = 1).

As discussed in 14.6., this interchange replaces Vi, v 2 by — v 2 , — Vi
and hence equally v #i /i, v, (/2 by — v #i/2 , — v^/i. Substituting these changes
into the above formulae (15:4), (15:5), it becomes clear that these formulae
must be modified only by replacing Max in them by Min. So we have:

(15:6)

(15:7)

v i = Min^ v #t /i,
v 2 = Min, v, /2 .

16 . 5 . 3 . We may sum up the formulae (15:2)-(15:7) of 15.4.2., 15.5.1.,
15.5.2., as follows:

, ai) define three

operations M k \> ki = 0, 1, 2 as follows:

(15:8)

Then

' 2 PibiWO

Max» i f(<r i)
Min, i f(<Ti)

v* = for

(= 1

f

, «i)

for

= 0 ,

for

ki

= 1,

for

ki

= 2.

k =

1,2.

We wish to emphasize some simple facts concerning these operations
M,\.

First, M k ,\ kills the variable a t ; i.e. M k \J(<n) no longer depends on o\.
For ki = 1, 2 — i.e. for Max, t , Min, i — this was pointed out in 13.2.3. For
ki = 0 it is obvious; and this operation is, by the way, analogous to the
integral used as an illustration in footnote 2 on p. 91.

Second, M k \ depends explicitly on the game T. This is evident since
k\ occurs in it and <t\ has the range 1, • • • , a\. But a further dependence
is due to the use of the pi(l), * • • , pi(«i), in the case of k\ = 0.

Third, the dependence of v* on v, /* is the same for k = 1, 2 for each
value of k\.

We conclude by observing that it would have been easy to make these
formulae — involving the average £ Pi(^i)/(^i) for a chance move, the

< r , « 1

maximum for a personal move of the first player, and the minimum for
one of his opponent — plausible by a purely verbal (unmathematical)
argument. It seemed nevertheless necessary to give an exact mathematical
treatment in order to do full justice to the precise position of Vi and of v 2 .
A purely verbal argument attempting this would unavoidably become so
involved — if not obscure — as to be of little value.

GAMES WITH PERFECT INFORMATION

123

16.6. The Result in the Case of Perfect Information

15 . 6 . 1 . We return now to the situation described at the end of 15.3.2.

and make all the hypotheses mentioned there; i.e. we assume that perfect
information prevails in the game Y and also that it is a zero-sum two-person
game. The scheme indicated loc. cit. , together with the formula (15:8) of
15.5.3. which takes care of the “inductive” step, enable us to determine the
essential properties of Y.

We prove first — without going any further into details — that such a T
is always strictly determined. We do this by “complete induction” with
respect to the length v of the game (cf. 15.1.2.). This consists of proving
two things :

(15:C:a) That this is true for all games of minimum length; i.e. for

v = 0.

(15:C:b) That if it is true for all games of length v — 1, for a given

v = 1, 2, • • • , then it is also true for all games of length v.

Proof of (15:C:a): If the length v is zero, then the game has no moves at
all; it consists of paying fixed amounts to the players 1, 2, — say the amounts
w, — w. 1 Hence = fh = 1 so ri = r 2 = 1, 3C(ri, r 2 ) = w, 2 and so

vi = v 2 = w;

i.e. T is strictly determined, and its v = w. 3

Proof of (15:C:b): Let Y be of length v. Then every r #i is of length
v — 1; hence by assumption every is strictly determined. Therefore
\ 9 /x == v #j/2 . Now the formula (15:8) of 15.5.3. shows 4 that Vi = v 2 .
Hence Y is also strictly determined and the proof is completed.

15 . 6 . 2 . We shall now go more into detail and determine the Vi = v 2 = v
of T explicitly. For this we do not even need the above result of 15.6.1.

We form, as at the end of 15.3.2., the sequence of games

(15:9)

r, r v r vv

, r, ,

1 r*V

of the respective lengths

v y v- 1, r - 2, • • • , 0.
Denote the Vi, v 2 of these games by

V*, y v *,. 9y/k'

1 Cf. the game in footnote 2 on p. 76 or in 15.3.1. In the partition and

set terminology: For v = 0 (10:1 :f), (10:1 :g) in 10.1.1. show that 12 has only one ele-
ment, say #: 12 = (w). So w = $i(w), — w = 3 2 (w) play the role indicated above.

* I.e. each player has only one strategy, which consists of doing nothing.

* This is, of course, rather obvious. The essential step is (15:C:b).

4 I.e. the fact mentioned at the end of 15.5.3., that the formula is the same for A; = 1,2
for each value of k\.

5 Cf. footnote 3 on p. 117.

124

ZERO-SUM TWO-PERSON GAMES: THEORY

Let us apply (15:8) of 15.5.3. for the “inductive” step described at

the end qf 15.3.2.; i.e. let us replace the <r h T, T^of 15.5.3. by<r„ r» t , Ki)

r,, for each k — 1, • • • , v. The k x of 15.5.3. then refers

to the first move of r, , ; i.e. to the move 9JI, in F. It is therefore

convenient to denote it by kjai, • • ■ , <r,-i). (Cf. 7.2.1.) Accordingly

we form the operation M k [ ( "' replacing the M k \ of 15.5.3. In this

way we obtain

(15:10)

v», ,

,/k

= M k /'"

• '.-l)

, a /k

for * = 1,2.

Consider now the last element of the sequence (15:9), the game V 9i v

This falls under the discussion of (15:C:a) in 15.6.1.; it has no moves at all.
Denote its unique play 1 by it = ir(<n, • • • , <r„). Hence its fixed w 2 is
equal to SFi(*(<ri, • • • , a,)). So we have:

(15:1 1) v^ *„/ 1 — v* 4 #„/2 SIiOKo’i, * • • , ov)).

Now apply (15:10) with k = v to (15:11) and then to the result, with
k = v — 1, • • • ,2, 1 successively. In this manner

(15:12) Vl = v s = v = M k 'M k \ ( ^ ■ • • M k /^ • • • , <r,)).

obtains.

This proves once more that T is strictly determined, and also gives an
explicit formula for its value.

15.7. Application to Chess

15 . 7 . 1 . The allusions of 6.4.1. and the assertions of 14.8. concerning
those zero-sum two-person games in which preliminarity and anteriority
coincide — i.e. where perfect information prevails — are now established. We
referred there to the general opinion that these games are of a particularly
rational character, and we have now given this vague view a precise meaning
by showing that the games In question are strictly determined. And we
have also shown — a fact much less founded on any “ general opinion” —
that this is also true when the game contains chance moves.

Examples of games with perfect information were already given in 6.4.1. :
Chess (without chance moves) and Backgammon (with chance moves).
Thus we have established for all these games the existence of a definite
value (of a play) and of definite best strategies. But we have established
their existence only in the abstract, while our method for their construction
is in most cases too lengthy for effective use. 3

In this connection it is worth while to consider Chess in a little more
detail.

1 Cf. the remarks concerning in 15.3.1.

* Cf . (15:C:a) in 15.6.1., particularly footnote 1 on p. 123.

* This is due mainly to the enormous value of v. For Chess, cf . the pertinent part of
footnote 3 on p. 59. (The v * there is our *, cf. the end of 7.2.3.)

GAMES WITH PERFECT INFORMATION

125

The outcome of a play in Chess — i.e. every value of the functions * of
6.2.2. or 9.2.4. — is restricted to the numbers 1, 0, — l. 1 Thus the functions
S* of 11.2.2. have the same values, and since there are no chance moves in
Chess, the same is true for the function 3C* of 11. 2.3. 2 In what follows we
shall use the functions = 3Ci of 14.1.1.

Sincere has only the values, 1, 0, —1, the number

(15:13) v = Max Ti Min Tt 5C(ri, r 2 ) = Min Tj Max Ti 3C(ri, t 2 )

has necessarily one of these values

v = 1, 0, -1.

We leave to the reader the simple discussion that (15:13) means this:

(15:D:a) If v = 1 then player 1 (“ white”) possesses a strategy with

which he “wins,” irrespective of what player 2 (“black”)
does.

(15:D:b) If v = 0 then both players possess a strategy with which

each one can “tie” (and possibly “win”), irrespective of
what the other player does.

(15:D:c) If v = —1 then player 2 (“black”) possesses a strategy

with which he “wins,” irrespective of what player 1 (“white”)
does. 8

»

15.7.2. This shows that if the theory of Chess were really fully known
there would be nothing left to play. The theory would show which of the
three possibilities (15:D:a), (15:D:b), (15:D:c) actually holds, and accord-
ingly the play would be decided before it starts : The decision would be in
case (15:D:a) for “white,” in case (15:D:b) for a “tie,” in case (15:D:c)
for “black.”

But our proof, which guarantees the validity of one (and only one)
of these three alternatives, gives no practically usable method to determine
the true one. This relative, human difficulty necessitates the use of those
incomplete, heuristic methods of playing, which constitute “good” Chess;
and without it there would be no element of “struggle” and “surprise” in
that game.

1 This is the simplest way to interpret a “win,” “tie,” or “loss” of a play by the
player k.

* Every value of §* is one of every value of 3C * — in the absence of chance moves — is
one of Qfc, cf. loc. cit. If there were chance moves, then the value of 3C* would be the
probability of a “win” minus that of a “loss,” i.e. a number which may lie anywhere
between 1 and —1.

* When there are chance moves, then 3C(ri, t 2 ) is the excess probability of a “win”
over a “loss,” cf. footnote 2 above. The players try to maximize or to minimize this
number, and the sharp trichotomy of (15:D:a) — (15:D:c) above does not, in general,
obtain.

Although Backgammon is a game in which complete information prevails, and which
contains chance moves, it is not a good example for the above possibility; Backgammon
is played for varying payments, and not for simple “win,” “tie” or “loss,” — i.e. the
values of the are not restricted to the numbers 1,0, — 1.

126

ZERO-SUM TWO-PERSON GAMES: THEORY

15.8. The Alternative, Verbal Discussion

16.8.1. We conclude this chapter by an alternative, simpler, less forma
istic approach to our main result, — that all zero-sum two-person games, i
which perfect information prevails, are strictly determined.

It can be questioned whether the argumentation which follows is reall
a proof; i.e., we prefer to formulate it as a plausibility argument by which
value can be ascribed to each play of any game T of the above type, but thi
is still open to criticism. It is not necessary to show in detail how thos
criticisms can be invalidated, since we obtain the same value v of a play c
T as in 15.4.-15.6., and there we gave an absolutely rigorous proc
using precisely defined concepts. The value of the present plausibilit
argument is that it is easier to grasp and that it may be repeated for othe
games, in which perfect information prevails, which are not subject to th
zero-sum two-person restriction. The point which we wish to bring ou
is that the same criticisms apply in the general case too, and that they ca
no longer be invalidated there. Indeed, the solution there will be foun
(even in games where perfect information prevails) along entirely differen
lines. This will make clearer the nature of the difference between th
zero-sum two-person case and the general case. That will be rathe
important for the justification of the fundamentally different method
which will* have to be used for the treatment of the general cas
(cf. 24.).

16.8.2. Consider a zero-sum two-person game T in which perfect informa
tion prevails. We use the notations of 15.6.2. in all respects: For th
9Tli, 9TC 2 , * * * , the a 1 , a 2f * * * , a 9 \ the k h k 2 (ai), * ' * , k p (cr Xy <r 2 , * * * , 0V-i)

the probabilities; the operators M k |, M k a \ iai \ • • • , ay ~ l) ; th

sequence (15:9) of games derived from r ; and the function • • • , a v ))

We proceed to discuss the game r by starting with the last move
and then going backward from there through the moves 9TC„_ 2 , * * *
Assume first that the choices a lf a 2 , * • • , <r 9 - \ (of the moves 3Tli, 3E 2 , • • •
9Tl,_i) have already been made, and that the choice <r ¥ (of the move 9TC„) i
now to be made.

If 9TC, is a chance move, i.e. if k p (a h a 2 , • • • , <r„_i) = 0, then a P wil
have the values 1, 2, • • • , a„(cri, • • • , <t„_i) with the respective probabili
ties p,(l), p*(2), • • • , p„(a„(<7i, • • • , o-„_i)). So the mathematical expec
tation of the final payment (for the player 1) ffi (*(<r lf • • • , cr „_ h <r„)) i

®r(*l. • • • . 0>-l)

2) • • • , ff.-i, <r .))•

" 1

If 9Tl„ is a personal move of players 1 or 2, i.e. if k ¥ {<r h * • • , o-„_ 1 ) =
or 2, then that player can be expected to maximize or to minimize
$i(ff(<ri, * * • , <x„_i, <r 9 )) by his choice of cr y ; i.e. the outcome
Max,, 2Fi(#(<r 1 , * * * , <r„_i, <r„)) or Min,^ 5i(*(<r h • • • , <r„_i, a,)), respec
tively is to be expected.

GAMES WITH PERFECT INFORMATION

127

I.e., the outcome to be expected for the play — after the choices ■ • • ,

<t,-\ have been made — is at any rate

M k f -’SF, Or(<n, • • • , a,)).

Assume next that only the choices m, • • • , a,_j (of the moves
91li, • • • , 3TC,_j) have been made and that the choice <r,_ i (of the move 9Il,_i)
is now to be made.

Since a definite choice of er._i entails, as we have seen, the outcome

*‘’' l) ffi(*(<ri, • , ir ,)) — which is a function of «n, • • • , <r,-i

only, since the operation m\\ (<r ‘ <r, ' l) kills a , — we can proceed as above.

We need only replace v; a, ■ • • , <r,; Si(w(a u • • • , <r,)) by

* - 1; *i, • • • , ■"-* • • • , a,)).

Consequently the outcome to be expected for the play — after the choices
<ri, • • • , <r „_2 have been made — is

"->'>M k ’?' "-*’SFi(ff (<r 1( • • • , <r,)).

Similarly the outcome to be expected for the play — after the choices
<t s have been made — is

''' _,) {Fi(»(«ri, • • • , a,)).

Finally, the outcome to be expected for the play outright — before it
has begun — is

M k \M k \ ■ ■ ■ M k ’-_'f' 1 ”" ) M k f' • • • , <r,)).

And this is precisely the v of (15:12) in 15.6.2. 1

15 . 8 . 3 . The objection against the procedure of 15.8.2. is that this
approach to the “value” of a play of r presupposes “rational” behavior
of all players; i.e. player Ts strategy is based upon the assumption that
player 2’s strategy is optimal and vice-versa.

Specifically: Assume • • • , (r,_ 2 ) = 1, k,{a h • • • , <r,-i) = 2.

Then player 1, whose personal move is 9TC„_i chooses his <r,_i in the convic-
tion that player 2, whose personal move is 9f chooses his <r„ “rationally.”
Indeed, this is his sole excuse for assuming that his choice of <r„_i entails

the outcome Min^SFiOrfo, • • • , o>)), i.e. * • • , <r,)),

of the play. (Cf. the discussion of 9TC„_i in 15.8.2.)

1 In imagining the application of this procedure to any specific game it must be remem-
bered that we assume the length v of r to be fixed. If v is actually variable — and it is
so in most games (cf. footnote 3 on p. 58) — then we must first make it constant, by
the device of adding “ dummy moves” to r as described at the end of 7.2.3. It is only
after this has been done that the above regression through 9Tl„ • • • , becomes

feasible.

For practical construction this procedure is of course no better than that of 15.4.-

15.6.

Possibly some very simple games, like Tit-tat-toe, could be effectively treated in
either manner.

128 ZERO-SUM TWO-PERSON GAMES: THEORY

Now in the second part of 4.1.2. we came to the conclusion that the
hypothesis of “ rationality ” in others must be avoided. The argumentation
of 15.8.2. did not meet this requirement.

It is possible to argue that in a zero-sum two-person game the rationality
of the opponent can be assumed, because the irrationality of his opponent
can never harm a player. Indeed, since there are only two players and
since the sum is zero, every loss which the opponent — irrationally — inflicts
upon himself, necessarily causes an equal gain to the other player. 1 As it
stands, this argument is far from complete, but it could be elaborated con-
siderably. However, we do not need to be concerned with its stringency:
We have the proof of 15.4.-15.6. which is not open to these criticisms. 2

But the above discussion is probably nevertheless significant for an
essential aspect of this matter. We shall see how it affects the modified
conditions in the more general case — not subject to the zero-sum two- person
restriction — referred to at the end of 15.8.1.

16. Linearity and Convexity
16.1. Geometrical Background

16.1.1. The task which confronts us next is that of finding a solution
which comprises all zero-sum two-person games, — i.e. which meets the
difficulties of the non-strictly determined case. We shall succeed in doing
this with the help of the same ideas with which we mastered the strictly
determined case : It will appear that they can be extended so as to cover all
zero-sum two-person games. In order to do this we shall have to make
use of certain possibilities of probability theory (cf. 17.1., 17.2.). And
it will be necessary to use some mathematical devices which are not quite
the usual ones. Our analysis of 13. provides one part of the tools; for the
remainder it will be most convenient to fall back on the mathematico-
geometrical theory of linearity and convexity. Two theorems on convex
bodies 3 will be particularly significant.

For these reasons we are now going to discuss — to the extent to which
they are needed — the concepts of linearity and convexity.

16 . 1 . 2 . It is not necessary for us to analyze in a fundamental way the
notion of n-dimensional linear (Euclidean) space. All we need to say is
that this space is described by n numerical coordinates. Accordingly we
define for each n = 1, 2, • • • , the n-dimensional linear space L n as the
set of all n-uplets of real numbers {x h • • • , x n }. These n-uplets can also
be looked upon as functions Xi of the variable i ) with the domain (1, • • • ,n)

1 This is not necessarily true if the sum is not constantly zero, or if there are more
than two players. For details cf. 20.1., 24.2.2., 58.3.

*Cf. in this respect particularly (14:D:a), (14:D:b), (14:C:d), (14:C:e) in 14.5.1.
and (14:C:a), U4:C:b) in 14.5.2.

1 Cf. T. Bonessen and W. Fenchel: Theorie der konvexen Korper, in Ergebnisse der
Mathematik und ihrer Grenzgebiete, Vol. III/l, Berlin 1934. Further investigations in
H. Weyl: Elementare Theorie der konvexen Polyeder. Commentarii Mathematici Helve-
tici, Vol. VII, 1935, pp. 290-306.

LINEARITY AND CONVEXITY

129

in the sense of 13.1.2., 13. 1.3. 1 We shall — in conformity with general
usage — call i an index and not a variable; but this does not alter the nature
of the case. In particular we have

{* 1 , •••,*»} = \yi, ■ ■ ■ , Vn)

if and only if Xi = i/< for all i = 1, • • • , n (cf. the end of 13.1.3.). One
could even take the view that L n is the simplest possible space of (numerical)
functions, where the domain is a fixed finite set — the set (1, • • • , n). 2

We shall also call these n-uplets — or functions — of L n points or vectors of
L„ and write

— >

(16:1) x = {*!, • • • , x„).

The Xi for the specific i = 1, • • • , n — the values of the function x, — are

— ^

the components of the vector x.

16 . 1 . 3 . We mention — although this is not essential for our further work
— that L n is not an abstract Euclidean space , but one in which a frame of
reference (system of coordinates) has already been chosen. 3 This is due
to the possibility of specifying the origin and the coordinate vectors of L n
numerically (cf. below) — but we do not propose to dwell upon this aspect
of the matter.

The zero vector or origin of L n is

0 - { 0 ,

The n coordinate vectors of L n are the

, 0 }.

, 1, • • • , 0} — { Si],

=

) &ni\ j — 1 ,

# = { 0 , •
where

i = j, 4 ' 5
i ^ j .

After these preliminaries we can now describe the fundamental oper-
ations and properties of vectors in L n .

for

for

16.2. Vector Operations

16 . 2 . 1 . The main operations involving vectors are those of scalar

multiplication j i.e. the multiplication of a vector x by a number t , and of

n.e. the n-uplets [ Xi, • • • , s„| are not merely sets in the sense of 8.2.1. The
effective enumeration of the x t by means of the index i = 1, • • • , n is just as essential
as the aggregate of their values. Cf . the similar situation in footnote 4 on p. 69.

* Much in modern analysis tends to corroborate this attitude.

8 This at least is the orthodox geometrical standpoint.

4 Thus the zero vector has all components 0, while the coordinate vectors have all
components but one 0 — that one component being 1, and its index j for thej-th coordinate
vector.

6 6ii is the “ symbol of Kronecker and Weierstrass,” which is quite useful in many
respects.

130

ZERO-SUM TWO-PERSON GAMES: THEORY

vector addition , i.e. addition of two vectors. The two operations are defined
by the corresponding operations, i.e. multiplication and addition, on the
components of the vector in question. More precisely:

Scalar multiplication: t{xi, • • • , x n ] = [tx i, • • • , tx n }.

Vector addition:

[X h ‘ ‘ , X n ) + \Vh ' * * , Vn] = {Xi + Vi, * * * , X n + Jfn 1 •

The algebra of these operations is so simple and obvious that we forego
its discussion. We note, however, that they permit the expression of any

vector x = {xi, * • • , x n \ with the help of its components and the coordi-
nate vectors of L n

x = ^ Xj 8 K 1
y»i

Some important subsets of L n :

(16:A:a) Consider a (linear, inhomogeneous) equation

n

(16:2:a) ^ a t Xi = b

t-i

(oi, • • • , a n , b are constants). We exclude
ai = • • • = a n — 0

since in that case there would be no equation at all. All

points (vectors) x = {xi, • • • , x n \ which fulfill this equation,
form a hyperplane . 2

(16:A:b) Given a hyperplane

(16:2:a) a x x { = 6,

»-i

it defines two parts of L n . It cuts L n into these two parts:

n

(16:2:b) % a,x,- > b,

»- 1

and

n

(16:2:c) ^ a^, < 6.

»-i

These are the two half-spaces produced by the hyperplane.

n

1 The x , are numbers, and hence they act in x, 8 ’ as scalar multipliers. ^ is a vector

y- 1

summation.

* For n ** 3, i.e. in ordinary (3-dimensional Euclidean) space, these are just the
ordinary (2-dimensional) planes. In our general case they are the ((n — 1 )-dimensional)
analogues; hence the name.

LINEARITY AND CONVEXITY

131

Observe that if we replace a i, • • • , a n , b by — a Xy • • , — a n , — 6, then
the hyperplane (16:2 :a) remains unaffected, but the two half-spaces (16:2:b),
(16:2:c) are interchanged. Hence we may always assume a half space to
be given in the form (16:2:b).

(16:A:c) Given two points (vectors) x , y and at ^ Owithl — i ^ 0;

then the center of gravity of y with the respective weights
t, 1 — t — in the sense of mechanics — is / x + (1 — t) y.

The equations

* • * , 2/n |,

, tX n (1 2/n }

X = {x h • • • , Xni, y = [y l, • • • , 2/n),

+(1—02/ = {tel + (1 — 02/i> * * * , ten + (1 — t)y n )
should make this amply clear.

A subset, C, of L n which contains all centers of gravity of all its points-

i.e. which contains with x , y M t x + (\ — t) y , 0 £ t £ 1 — is convex.

The reader will note that for n = 2,

3 — i.e. in the ordinary plane or space — ,

this is the customary concept of con- wTTTT/// / //// /// nK

vexity. Indeed, the set of all points /yy/A

tx + (l — 0 2/ > 0 = * =£ 1 is precisely / f //////// . // ////w ^

the linear (straight) interval connect- K/y/y^^

ing the points x and y , the interval

[ x , t/ ]. And so a convex set is one vZ /y/w

which, with any two of its points x , interval ix

Interval | j y |

Figure 10.

y , also contains their interval [ x , y ].

Figure 16 shows the conditions for n — 2, i.e. in the plane.

16 . 2 . 2 . Clearly the intersection of any number of convex sets is again

convex. Hence if any number of points (vectors) x x p is given,

there exists a smallest convex set containing them all: the intersection of

— ► — ►

all convex sets which contain x ',*••, x p . This is the convex set spanned
— ► — ►

by x ',*•*, x p . It is again useful to visualize the case n = 2 (plane).
Cf. Fig. 17, where p = 6. It is easy to verify that this set consists of all
points (vectors)

9 p p

(16:2:d) ^ tj x * for all t\ ^ 0, • • • , t p ^ 0 with = 1.

;-i j=i

— ^ ^

Proof: The points (16:2:d) form a set containing all x • • • x p .

— ►

x ’ is such a point: put tj = 1 and all other t* = 0.

132

ZERO-SUM TWO-PERSON GAMES: THEORY

The points (16:2 :d) form a convex set: If x = 2 ) J/^'and y = £ a/x*,

i-i ;-i

then J x + (1 — t) y = ^ u i x i with Uj = ttj + (1 — <)$y.

y-i

— > — >

Any convex set, D, containing x x p contains also all points

of (16:2:d): We prove this by induction for all p = 1, 2, • • • .

Proof: For p — 1 it is obvious; since then h = 1 and so x ' is the only
point of (16:2:d).

p - 1

Assume that it is true f or p — 1 . Consider p itself. If £ = 0 then

y-i

<!*•••= < P _i = 0, the point of (16:2:d) is x p and thus belongs to D. If

Figure 17.

p— l p— 1 p p— l

5) <, > 0, then put t = £ so 1 — t = ]>) *, — £ = <p. HenceO < < ^ 1.

i - i /-i i-i

P-1

Put a,- = (,•/< for j = 1, • • • , p — 1. So £ s j — 1- Hence, by our

p-i

assumption for p — 1, a, x * is in D.

i-i

y-i

D is convex, hence

p- 1 _» — ►

t X «,•*' + (l — 0
y-i

is also in Z); but this vector is equal to

p— 1 p — ►

2) tjx> + (,*>’ = 2)

/-i i-i

which thus belongs to D.

LINEARITY AND CONVEXITY

133

The proof is therefore completed.

The < 1 , • • • , t p of (16:2:d) may themselves be viewed as the com-
— ►

ponents of a vector t = {$ 1 , * * * , t p ) inL p . It is therefore appropriate
to give a name to the set to which they are restricted, defined by

tikO, • • • , t p £ 0,

and

i t,- = i.

y - 1

Figure 18. Figure 19.

Figure 20. Figure 21.

We shall denote it by S p . It is also convenient to give a name to the set
which is described by the first line of conditions above alone, i.e. by t\ ^ 0,
• • • , t p ^ 0. We shall denote it by P p . Both sets S P) P p are convex.

Let us picture the cases p = 2 (plane) and p = 3 (space). P 2 is the
positive quadrant , the area between the positive x\ and X 2 axes (Figure 18).
Pa is the positive octant , the space between the positive x\ } x% and x s axes, —
i.e. between the plane quadrants limited by the pairs Xi, x 2 ; x h x 8 ; x 2 , x 2 of
these (Fig. 19). /S 2 is a linear interval crossing P 2 (Figure 18). S* is a plane

triangle, likewise crossing Pa (Figure 19). It is useful to draw Sty S% sep-

134

ZERO-SUM TWO-PERSON GAMES: THEORY

arately, without the Pa, P 3 (or even the La, L 8 ) into which they are naturally
immersed (Figures 20, 21). We have indicated on these figures those dis-
tances which are proportional to x h x 2 or xi, x 2) x 8 , respectively.

(We re-emphasize: The distances marked x h x 2 , x 8 in Figures 20, 21 are
not the coordinates x h x 2 , x 8 themselves. These lie in La or L 8 outside of
S* or Sty and therefore cannot be pictured in S 2 or S 3 ; but they are easily
seen to be proportional to those coordinates.)

16 . 2 . 3 . Another important notion is the length of a vector. The length

of x = {xi, • • • , x n J is

The distance of two points (vectors) is the length of their difference:

- y\ = J X (*< - »<)*

* »« 1

Thus the length of x is the distance from the origin 0 .*

16.3. The Theorem of the Supporting Hyperplanes
16 . 3 . We shall now establish an important general property of convex
sets:

(16:B) Let p vectors x ! , • • • , x p be given. Then a vector y

— > — >

either belongs to the convex C spanned by x ', • * , x p (cf.

(16:A:c) in 16.2.1.), or there exists a hyperplane which contains

y (cf. (16:2:a) in 16.2.1.) such that all of C is contained in
one half-space produced by that hyperplane (say (16:2:b) in
16.2.1.; cf. (16:A:b) id.).

This is true even if the convex spanned by x ',••*, x p is replaced by
any convex set. In this form it is a fundamental tool in the modern theory
of convex sets.

A picture in the case n = 2 (plane) follows: Figure 22 uses the convex
set C of Figure 17 (which is spanned by a finite number of points, as in
the assertion above), while Figure 23 shows a general convex set C. 1 2

Before proving (16:B), we observe that the second alternative clearly
— ►

excludes the first, since y belongs to the hyperplane, hence not to the half
space. (I.e. it fulfills (16:2:a) and not (16:2:b) in (16:A:b) above.)

We now give the proof:

Proof: Assume that y does not belong to C. Then consider a point of
— >

C which lies as near to y as possible, — i.e. for which

1 The Euclidean — Pythagorean — meaning of these notions is immediate.

*For the reader who is familiar with topology, we add: To be exact, this sentence
should be qualified — the statement is meant for closed convex sets. This guarantees
the existence of the minimum that we use in the proof that follows. Regarding these

LINEARITY AND CONVEXITY

135

1 2 - y I 2 = 2 (*• - y<y

t-l

assumes its minimum value.

Consider any other points u of C. Then for every t with 0 t 5* 1,

t u + (1 — t) z also belongs to the convex C. By virtue of the minimum
— ►

property of z (cf. above) this necessitates

I tu + (1 - t) z - y | 2 ^ | z - y | 2 ,

i.e.

i.e.

I (z - y) +t(u - z) | 2 ^ | z - y | 2 ,

X !(*• - 2/0 + <( w «- - 2 t )) 2 ^ - </») 2 -

»-i

t-i

By elementary algebra this means

2 £ — 2/0 (w» ~ *0* + 2) (w» ~ s») 2 * 2 ^ 0.

»-i t-i

136 ZERO-SUM TWO-PERSON GAMES: THEORY

So for t > 0 (but of course t S 1) even

2 X (*,- - yi)(Ui - 2 .) + £ («< - *<)*< ^ 0.

t-1 t-1

n

If t converges toO, then the left-hand side converges to 2 £ (s» — — *,•).

t-i

Hence

(16:3)

X & ~ - *0 ^ °-

»-i

As — yi = (w< — 2 t ) + ( 2 * — y»), this means

X - »•)(«< - y.) ^ X ( Zi - y<) 2 = 1 2 - 1 / 1*-

t-1

t-1

Now z 7 * y (as z belongs to C, but y does not); hence | z — y | 2 > 0.
So the left-hand side above is > 0. I.e.

(16:4) X ( z< “ V‘)«i > X ( Zi "

t-1 t-1

Put di ~ z% — y, then ai = • • • = a n = 0 is excluded by z j* y (cf.

n

above). Put also b — ^ a#,. Thus

t-i

n

(16:2 :a*) £ aan = &

t-1

defines a hyperplane, to which y clearly be-
longs. Next

n

(16:2:b*) £ a <*,- > b

»- 1

is a half space produced by this hyperplane, and (16:4) states precisely

that u belongs to this half space.

Since u was an arbitrary element of C this completes the proof.

This algebraic proof can also be stated in the geometrical language.

Let us do this for the case n — 2 (plane) first. The situation is pictured

in Figure 24: z is a point of C which is as near to the given point y as possible;

i.e. for which the distance of y and z ,\ z — y | assumes its minimum value.

LINEARITY AND CONVEXITY

137

Since y } z are fixed, and u is a variable point (of C), therefore (16:3) defined
a hyperplane and one of the half spaces produced by it. And it is easy

to verify that z belongs to this hyperplane, and that it consists of those
— ►

points u for which the angle formed by the three points is a right-angle

(i.e. for which the vectors z — y and u - z are orthogonal). This means,

indeed, that £ (z< — yi)(ui — z* ) = 0. Clearly all of C must lie on this

»-i

hyperplane, or on that side of it which is away from y . If any point u

of C did lie on the y side, then some points of the interval [ z , u ] would be
— > — ►

nearer to y than z is. (Cf. Figure 25. The computation on pp. 135-136 —
properly interpreted — shows precisely this.) Since C contains z and u ,
and so all of [ z , u ], this would contradict the statement that z is as near
to y as possible in C.

Now our passage from (16:3) to (16:4) amounts to a parallel shift of
this hyperplane from z to y (parallel, because the coefficients a t =

138

ZERO-SUM TWO-PERSON GAMES: THEORY

of Ui, i = 1, • • • , n are unaltered). Now y lies on the hyperplane, and
all of C in one half-space produced by it (Figure 26).

The case n = 3 (space) could be visualized in a similar way.

It is even possible to account for a general n in this geometrical manner.
If the reader can persuade himself that he possesses n-dimensional “ geo-
metrical intuition” he may accept the above as a proof which is equally
valid in n dimensions. It is even possible to avoid this by arguing as

follows: Whatever n, the entire proof deals with only three points at once,
— ► - — > — ►

namely y , z , u . Now it is always possible to lay a (2-dimensional) plane
through three given points. If we consider only the situation in this
plane, then Figures 24-26 and the associated argument can be used without
any re-interpretation.

Be this as it may, the purely algebraic proof given above is absolutely
rigorous at any rate. We gave the geometrical analogies mainly in the
hope that they may facilitate the understanding of the algebraic operations
performed in that proof.

16.4. The Theorem of the Alternative for Matrices

16 . 4 . 1 . The theorem (16:B) of 16.3. permits an inference which will be
fundamental for our subsequent work.

We start by considering a rectangular matrix in the sense of 13.3.3.
with n rows and m columns, and the matrix element a(i } j). (Cf. Figure 11
in 13.3.3. The <t>, x , y , t, s there correspond to our a, i, j, n, m.) I.e.
a(t, j) is a perfectly arbitrary function of the two variables i = 1, • • • , n;

j = 1, • • • , m. Next we form certain vectors in L n : For each j = 1, • • • ,
— ►

m the vector x i — {x\, • • • , with x{ = a(i f j) and for each l = 1,

• • • , n the coordinate vector 8 1 = {8 l i\. (Cf. for the latter the end of
16.1.3.; we have replaced the j there by our l.) Let us now apply the

theorem (16 :B) of 16.3. for p = n + m to these n + m vectors x ', • • • ,

x m , 8 ', • • • , 8 n . (They replace the x ',*•*, x p loc. cit.) We put

y = 0 .

The convex C spanned by x ', • • •

, 8 n may contain 0 .

If this is the case, then we can conclude from (16 :2 :d) in 16.2.2. that

£ tiXj + 2 ' = 0 ,

i = i

with

(16:5) <i 0, ^ 0, ^ 0, 0.

m n

(16:6) X l > + t s < = L

i-1 i-l

LINEARITY AND CONVEXITY 139

ti, * * • ,t m ,s i, • • • , $ n replace the t h • • • , ^ (loc. cit.). In terms of the
components this means

X *<«(*’» i) + 2) = 0-

j-i /-i

The second term on the left-hand side is equal to a<, so we write

m

(16:7) £ a(i, j)* ; = -s t .

y- 1

If we had U = 0, then ^ = • • • = * m = 0, hence by (16:7) $!=••• =
y-i

m

s n = 0, thus contradicting (16:6). Hence ^ > 0. We replace (16:7)

y-i

by its corollary
(16:8)

X «(*» i)<; ^ o.

y- 1

/ m m

^ t, for j = 1, • • ■ , m. Then we have ^ x, = 1 and

y-i

(16:5) gives x x ^ 0, • * • , x m ^ 0. Hence

(16:9) x = {xi, • • • , x m \ belongs to S„

and (16:8) gives

(16:10)

X a(i, j)xj ^ 0 for * = !,•••

n.

i-i

Consider, on the other hand, the possibility that C does not contain 0 .
Then the theorem (16:B) of 16.3. permits us to infer the existence of a

hyperplane which contains y (cf. (16:2:a) in 16.2.1.), such that all of C
is contained in one half-space produced by that hyperplane (cf. (16:2:b) in
16.2.1.). Denote this hyperplane by

5) dxXi = 6 .

t = i

— ►

Since 0 belongs to it, therefore 6 = 0. So the half space in question is

n

X a t Xi > 0 .

i-l

(16:11)

140

ZERO-SUM TWO-PERSON GAMES: THEORY

x x ", 5 • • • , 6 " belong to this half space. Stating this for

v n

8 (16:11) becomes £ o,5,i > 0, i.e. a< > 0. So we have

(16:12) oj > 0, • • • , a n > 0.

Stating it for x (16:11) becomes

(16:13) 2a(i,j)a<> 0.

t-1

Now put Wi

a x for i = !,•••

, n.

(16:12) gives w\ > 0, • • * , w n > 0. Hence

n

Then we have £ u>, = 1 and

t-i

(16:14) w = (toi, • • • , win) belongs to S„.

And (16:13) gives

m

(16:15) £ a(i,j)Wi 0 for j = 1, • • • , m.

t - 1

Summing up (16:9), (16:10), (16:14), (16:15), we may state:

(16 :C) Let a rectangular matrix with n rows and m columns be

given. Denote its matrix element by a(i , j), i = 1, * • • , n;

— ►

j = m. Then there exists either a vector x —

{xi, • • ■ , x m ) in S m with

m

(16:16:a) 2} a (hJ) x i ^ 0 for i = 1, * * * , n,

y-i

or a vector it? = {it?i, • • • , it? n ) in S n with

n

(16:16:b) 2} a (h j) w < >0 for j = 1, • * , m.

»-i

We observe further:

The two alternatives (16:16:a), (16:16:b) exclude each other.

Proof: Assume both ( 1 6 : 1 6 :a) and ( 1 6 : 1 6 :b) . Multiply each ( 1 6 : 1 6 :a) by

n m

W{ and sum over i = 1, • • • , n; this gives £ £ a(i,j)wiXj£0. Multiply

i-u-i

LINEARITY AND CONVEXITY

141

each (16:16:b) by x, and sum over j = 1, • • • , m; this gives

£ £ a (h j)wiXj > 0. 1

Thus we have a contradiction.

16.4.2. We replace the matrix o(i, j) by its negative transposed matrix;
i.e. we denote the columns (and not, as before, the rows) by i = 1 , • • • , n
and the rows (and not, as before, the columns) by j = 1, • • • , ra. And
we let the matrix element be — a(i, j) (and not, as before a(z, j)). (Thus
n, m too are interchanged.)

We restate now the final results of 16.4.1. as applied to this new matrix.

But in formulating them, we let x ' = {x[, * • • , x' m ] play the role which

w = {w h • * • , w n \ had, and w' = {w[, • * * , w' n \ the role which x ^
{xi, • • * , x m ) had. And we announce the result in terms of the original
matrix.

Then we have:

(16 :D) Let a rectangular matrix with n rows and m columns be

given. Denote its matrix element by a(i y j), i = 1, • • • , n;

— ►

j = 1, • ■ • , m. Then there exists either a vector x ' =
\x\, • • • , x'J in S m with

m

(16:17 :a) £ a(i, j)x' <0 for i = 1, • • • , n,

y-i

or a vector w;' = {u/ L , • • • , w'J in S n with

n

(16:17 :b) ^ a(i, j)Wf ^0 for j = 1, • • • , m.

1-1

And the two alternatives exclude each other.

16 . 4 . 3 . We now combine the results of 16.4.1. and 16.4.2. They imply
that we must have (16:17:a), or (16:16 :b), or (16:16:a) and (16:17:b)
simultaneously; and also that these three possibilities exclude each other.

Using the same matrix a(i , j) but writing x , w , x \ w' for the vectors

x w , x , w ' in 16.4.1., 16.4.2. we obtain this:

1 > 0 and not only ^ 0. Indeed, * 0 would necessitate x x = 0 which

m

it impossible since ^ x, * 1.

i-i

142 ZERO-SUM TWO-PERSON GAMES: THEORY

(16:E) There exists either a vector x = }xi, • • • , x m \ in S m with

m

(16:18:a) 2} a(i, j)x ,• <0 for i = 1, • • • , n,

;-i

or a vector w = [w\> • • • , w n } in S n with

n

(.16:18 :b)

X > 0

1

for

j = 1, • • • , m,

or two vectors x ' = {
in S n with

X \y

— ►

, x' m \ inland w’ =

<s> .

IIA

o

for

*' = 1, • • • , n,

(16:18:c)

i-l

n

TTjXl

<s» .

<-o.

A

IIV

o

for

j = 1, •• • , m.

The three alternatives (16:18:a), (16:8:b), (16:8:c) exclude each
other.

By combining (16:18 :a) and (16:18 :c) on one hand and (16:18 :b) and
(16:18:c) on the other, we get this simpler but weaker statement. 12

(16:F) There exists either a vector x = {xi, • • • , x m } in S m with

(16:19:a) X a ( z > 3) x i = 0 for i = 1, • • • , n,
y- 1

or a vector w = {wi, * * * , w n \ in S n with

(16:19 :b) 2) a (h j) w * = 0 for j = 1, * • • , m.

i-1

16 . 4 . 4 . Consider now a skew symmetric matrix a(i, j), i.e. one which
coincides with its negative transposed in the sense of 16.4.2. ; i.e. n = m and

j) = -<* 0 \ 0 for i, j = 1, • • • , n.

1 The two alternatives (16:19:a), (16:19:b) do not exclude each other: Their conjunc-
tion is precisely (16:18:c).

8 This result could also have been obtained directly from the final result of 16.4.1.:
(16:19:a) is precisely (16:16:a) there, and (16:19:b) is a weakened form of (16:16:b)
there. We gave the above more detailed discussion because it gives a better insight
into the entire situation.

MIXED STRATEGIES. THE SOLUTION

143

Then the conditions (16:19:a) and (16:19 :b) in 16.4.3. express the same
thing: Indeed, (16:19:b) is

n

X a(i,j)Wi ^ 0;

t-1

this may be written

n n

“ £ a(j, * 0, or £ a(j, i)wi £ 0.

»-i *-i

n

We need only write j, i for i, j 1 so that this becomes £ o(*, ^ 0, and

i-i

then x for w , l so that we have ^ a(i, j)xj ^ 0. And this is precisely

;»i

(16:19:a).

Therefore we can replace the disjunction of (16:19:a) and (16:19 :b)
by either one of them, — say by (16:19 :b). So we obtain:

(16 :G) If the matrix a(i, j ) is skew-symmetric (and therefore n = m

cf. above), then there exists a vector w = [wi, • • • , to n } in
S n with

n

^ a{i, j)wi ^0 for j = 1, •• • , n.

»- 1

17. Mixed Strategies. The Solution for All Games

17.1. Discussion of Two Elementary Examples

17.1.1. In order to overcome the difficulties in the non-strictly determined
case — which we observed particularly in 14.7. — it is best to reconsider the
simplest examples of this phenomenon. These are the games of Matching
Pennies and of Stone, Paper, Scissors (cf. 14.7.2., 14.7.3.). Since an
empirical, common-sense attitude with respect to the “problems” of these
games exists, we may hope to get a clue for the solution of non-strictly
determined (zero-sum two-person) games by observing and analyzing these
attitudes.

It was pointed out that, e.g. in Matching Pennies, no particular way
of playing — i.e. neither playing “heads” nor playing “tails” — is any
better than the other, and all that matters is to find out the opponent^ inten-
tions. This seems to block the way to a solution, since the rules of the
game in question explicitly bar each player from the knowledge about the
opponent’s actions, at the moment when he has to make his choice. But

1 Observe that now, with n = m this is only a change in notation !

144 ZERO SUM TWO-PERSON GAMES: THEORY

the above observation does not correspond fully to the realities of the case:
In playing Matching Pennies against an at least moderately intelligent
opponent, the player will not attempt to find out the opponent's intentions
but will concentrate on avoiding having his own intentions found out, by
playing irregularly “ heads" and “ tails" in successive games. Since we
wish to describe the strategy in one play — indeed we must discuss the course
in one play and not that of a sequence of successive plays — it is preferable
to express this as follows : The player's strategy consists neither of playing
“tails” nor of playing “heads,” but of playing “tails” with the probability
of i and “heads” with the probability of i.

17.1.2. One might imagine that in order to play Matching Pennies in a
rational way the player will — before his choice in each play — decide by
some 50:50 chance device whether to play “heads” or “tails.” 1 The
point is that this procedure protects him from loss. Indeed, whatever
strategy the opponent follows, the player's expectation for the outcome
of the play will be zero. 2 This is true in particular if with certainty the
opponent plays “tails,” and also if with certainty he plays “heads”; and
also, finally, if he — like the player himself — may play both “heads” and
“tails,” with certain probabilities. 3

Thus, if we permit a player in Matching Pennies to use a “statistical”
strategy, i.e. to “mix” the possible ways of playing with certain proba-
bilities (chosen by him), then he can protect himself against loss. Indeed,
we specified above such a statistical strategy with which he cannot lose,
irrespective of what his opponent does. The same is true for the opponent,
i.e. the opponent can use a statistical strategy which prevents the player
from winning, irrespective of what the player does. 4

The reader will observe the great similarity of this with the discussions of
14.5. 5 In the spirit of those discussions it seems legitimate to consider
zero as the value of a play of Matching Pennies and the 50 : 50 statistical
mixture of “heads” and “tails” as a good strategy.

The situation in Paper, Stone, Scissors is entirely similar. Common
sense will tell that the good way of playing is to play all three alternatives
with the probabilities of £ each. 6 The value of a play as well as the inter-

1 E.g. he could throw a die — of course without letting the opponent see the result —
and then play “tails” if the number of spots showing is even, and “heads” if that num-
ber is odd.

• I.e. his probability of winning equals his probability of losing, because under these
conditions the probability of matching as well as that of not matching will be J, what-
ever the opponent's conduct.

8 Say p y 1 — p. For the player himself we used the probabilities J, i.

4 All this, of course, in the statistical sense: that the player cannot lose, means that
his probability of losing is ^ his probability of winning. That he cannot win, means
that the former is to the latter. Actually each play will be won or lost, since Matching
Pennies knows no ties.

• We mean specifically (14:C:d), (14:C:e) in 14.5.1.

• A chance device could be introduced as before. The die mentioned in footnote 1,
above, would be a possible one. E.g. the player could decide “stone” if 1 or 2 spots
show, “paper” is 3 or 4 spots show, “scissors” if 5 or 6 show.

MIXED STRATEGIES. THE SOLUTION

145

pretation of the above strategy as a good one can be motivated as before,
again in the sense of the quotation there, 1

17.2. Generalization of This Viewpoint

17.2.1. It is plausible to try to extend the results found for Matching
Pennies and Stone, Paper, Scissors to all zero-sum two-person games.

We use the normalized form, the possible choices of the two players
being n = 1, • • • , 0i and t 2 = 1, * • * , p 2 , and the outcome for player 1
3C(ri, t 2 ), as formerly. We make no assumption of strict determinateness.

Let us now try to repeat the procedure which was successful in 17.1.; i.e.
let us again visualize players whose “theory” of the game consists not in
the choice of definite strategies but rather in the choice of several strategies
with definite probabilities. 2 Thus player 1 will not choose a number
n = 1, • • • , pi — i.e. the corresponding strategy — but Pi numbers

fi, • • • , — the probabilities of these strategies 2)J, • • * , 2?*, respec-
tively. Equally player 2 will not choose a number r 2 = 1, • • • , Pi — i.e.
the corresponding strategy 2^* — but p 2 numbers iji, • • • , 17^ — the proba-
bilities of these strategies 2' 2 , • • • , 2^», respectively. Since these prob-
abilities belong to disjoint but exhaustive alternatives, the numbers £ Tl > tj Tj
are subject to the conditions

(17:1 :a) all ^ £ 0, £ - 1;

Tf-1

(17:1 :b) allifr.fcO, f Vr, = 1.

r* ” 1

and to no others.

— ► — >

We form the vectors ( = {£1, • • • , and rj = { rj lf • • • , rj^J.

— > — ^

Then the above conditions state that { must belong to S/ 3 t and rj to
in the sense of 16.2.2.

In this setup a player does not, as previously, choose his strategy, but
he plays all possible strategies and chooses only the probabilities with which
he is going to play them respectively. This generalization meets the major
difficulty of the not strictly determined case to a certain point: We have
seen that the characteristic of that case was that it constituted a definite
disadvantage 3 for each player to have his intentions found out by his

1 In Stone, Paper, Scissors there exists a tie, but no loss still means that the probability
of losing is 25 the probability of winning, and no gain means the reverse. Cf. footnote
4 on p. 144.

* That these probabilities were the same for all strategies ( J, i or i, J, i in the exam-
ples of the last paragraph) was, of course accidental. It is to be expected that this
equality was due to the symmetric way in which the various alternatives appeared in
those games. We proceed now on the assumption that the appearance of probabiU|ie8
in formulating a strategy was the essential thing, while the particular values* were
accidental.

•The A > Oof 14.7.1.

146

ZERO-SUM TWO-PERSON GAMES: THEORY

opponent. Thus one important consideration 1 for a player in such a game
is to protect himself against having his intentions found out by his opponent.
Playing several different strategies at random, so that only their probabili-
ties are determined, is a very effective way to achieve a degree of such
protection: By this device the opponent cannot possibly find out what the
player's strategy is going to be, since the player does not know it himself. 2
Ignorance is obviously a very good safeguard against disclosing information
directly or indirectly.

17 . 2 . 2 . It may now seem that we have incidentally restricted the
player's freedom of action. It may happen, after all, that he wishes to
play one definite strategy to the exclusion of all others; or that, while desir-
ing to use certain strategies with certain probabilities, he wants to exclude
absolutely the remaining ones. 3 We emphasize that these possibilities are
perfectly within the scope of our scheme. A player who does not wish to
play certain strategies at all will simply choose for them the probabilities
zero. A player who wishes to play one strategy to the exclusion of all
others will choose for this strategy the probability 1 and for all other
strategies the probability zero.

Thus if player 1 wishes to play the strategy only, he will choose for £
the coordinate vector 5 T i (cf. 16.1.3.). Similarly for player 2, the strategy
and the vectors r\ and 8 T *.

In view of all these considerations we call a vector £ of or a vector

rj of Sp t a statistical or mixed strategy of player 1 or 2, respectively. The
— ► — >

coordinate vectors 8 T 1 or 8 r * correspond, as we saw, to the original strategies
r 1 or r 2 — i.e. or X r 2 * — of player 1 or 2, respectively. We call them strict

or pure strategies .

17.3. Justification of the Procedure As Applied to an Individual Play

17 . 3 . 1 . At this stage the reader may have become uneasy and perceive
a contradiction between two viewpoints which we have stressed as equally
vital throughout our discussions. On the one hand we have always insisted
that our theory is a static one (cf. 4.8.2.), and that we analyze the course

1 But not necessarily the only one.

2 If the opponent has enough statistical experience about the player’s “style,” or if
he is very shrewd in rationalizing his expected behavior, he may discover the probabilities
— frequencies — of the various strategies. (We need not discuss whether and how this
may happen. Cf. the argument of 17.3.1.) But by the very concept of probability
and randomness nobody under any conditions can foresee what will actually happen in
any particular case. (Exception must be made for such probabilities as may vanish;
cf. below.)

3 In this case he clearly increases the danger of having his strategy found out by the
opponent. But it may be that the strategy or strategies in question have such intrinsic
advantages over the others as to make this worth while. This happens, — e.g. in an
extreme form for the “good” strategies of the strictly determined case (cf. 14.5., particu-
larly (14:C:a), (14:C:b) in 14.5.2.).

MIXED STRATEGIES. THE SOLUTION

147

of one play and not that of a sequence of successive plays (cf. 17.1.). But
on the other hand we have placed considerations concerning the danger of
one’s strategy being found out by the opponent into an absolutely central
position (cf. 14.4., 14.7.1. and again the last part of 17.2.). How can
the strategy of a player — particularly one who plays a random mixture of
several different strategies — be found out if not by continued observation!
We have mled out that this observation should extend over many plays.
Thus it would seem necessary to carry it out in a single play. Now even
if the rules of the game should be such as to make this possible — i.e. if they
lead to long and repetitious plays — the observation would be effected only
gradually and successively in the course of the play. It would not be avail-
able at the beginning. And the whole thing would be tied up with various
dynamical considerations, — while we insisted on a static theory ! Besides,
the rules of the game may not even give such opportunities for observation; 1
they certainly do not in our original examples of Matching Pennies, and
Stone, Paper, Scissors. These conflicts and contradictions occur both in the
discussions of 14. — where we used no probabilities in connection with the
choice of a strategy — and in our present discussions of 17. where probabilities
will be used.

How are they to be solved?

17.3.2. Our answer is this :

To begin with, the ultimate proof of the results obtained in 14. and 17. —
i.e. the discussions of 14.5. and of 17.8. — do not contain any of these con-
flicting elements. So we could answer that our final proofs are correct
even though the heuristic procedures which lead to them are questionable.

But even these procedures can be justified. We make no concessions:
Our viewpoint is static and we are analyzing only a single play. We are
trying to find a satisfactory theory, — at ‘this stage for the zero-sum two-
person game. Consequently we are not arguing deductively from the firm
basis of an existing theory — which has already stood all reasonable tests —
but we are searching for such a theory. 2 Now in doing this, it is perfectly
legitimate for us to use the conventional tools of logics, and in particular
that of the indirect proof. This consists in imagining that we have a satis-
factory theory of a certain desired type, 3 trying to picture the consequences
of this imaginary intellectual situation, and then in drawing conclusions
from this as to what the hypothetical theory must be like in detail. If this
process is applied successfully, it may narrow the possibilities for the hypo-
thetical theory of the type in question to such an extent that only one

1 I.e. “gradual,” “successive” observations of the behavior of the opponent within
one play.

* Our method is, of course, the empirical one: We are trying to understand, formalize
and generalize those features of the simplest games which impress us as typical. This is,
after all, the standard method of all sciences with an empirical basis.

3 This is full cognizance of the fact that we do not (yet) possess one, and that we
cannot imagine (yet) what it would be like, if we had one.

All this is — in its own domain — no worse than any other indirect proof in any part of
science (e.g. the per absurdum proofs in mathematics and in physics).

148 ZERO-SUM TWO-PERSON GAMES: THEORY

possibility is left, — i.e. that the theory is determined, discovered by this
device. 1 * * Of course, it can happen that the application is even more u suc-
cessful, and that it narrows the possibilities down to nothing — i.e. that it
demonstrates that a consistent theory of the desired kind is inconceivable.*

17 . 3 . 3 . Let us now imagine that there exists a complete theory of the
zero-sum two-person game which tells a player what to do, and which is
absolutely convincing. If the players knew such a theory then each player
would have to assume that his strategy has been “ found out” by his oppo-
nent. The opponent knows the theory, and he knows that a player would be
unwise not to follow it. 8 Thus the hypothesis of the existence of a satis-
factory theory legitimatizes our investigation of the situation when a play-
er’s strategy is “ found out” by his opponent. And a satisfactory theory 4 *
can exist only if we are able to harmonize the two extremes Ti and r 2 , —
strategies of player 1 “ found out” or of player 2 “ found out.”

For the original treatment — free from probability (i.e. with pure strate-
gies) — the extent to which this can be done was determined in 14.5.

We saw that the strictly determined case is the one where there exists a
theory satisfactory on that basis. We are now trying to push further, by
using probabilities (i.e. with mixed strategies). The same device which
we used in 14.5. when there were no probabilities will do again, — the analysis
of “ finding out” the strategy of the other player.

It will turn out that this time the hypothetical theory can be determined
completely and in all cases (not merely for the strictly determined case —
cf. 17.5.1., 17.6.).

After the theory is found we must justify it independently by a direct
argument . 6 * This was done for the strictly determined case in 14.5., and
we shall do it for the present complete theory in 17.8.

1 There are several important examples of this performance in physics. The succes-
sive approaches to Special and to General Relativity or to Wave Mechanics may be
viewed as such. Cf. A. D’Abro: The Decline of Mechanism in Modern Physics, New
York 1939.

* This too occurs in physics. The N. Bohr-Heisenberg analysis of “quantities which
are not simultaneously observable” in Quantum Mechanics permits this interpretation.
Cf. N . Bohr: Atomic Theory and the Description of Nature, Cambridge 1934 and P. A. M.
Dirac : The Principles of Quantum Mechanics, London 1931, Chap. I.

8 Why it would be unwise not to follow it is none of our concern at present; we have
assumed that the theory is absolutely convincing.

That this is not impossible will appear from our final result. We shall find a theory
which is satisfactory; nevertheless it implies that the player's strategy is found out by his
opponent. But the theory gives him the directions which permit him to adjust himself
so that this causes no loss. (Cf. the theorem of 17.6. and the discussion of our complete
solution in 17.8.)

4 I.e. a theory using our present devices only. Of course we do not pretend to be
able to make “absolute” statements. If our present requirements should turn out to be
unfulfillable we should have to look for another basis for a theory. We have actually

done this once by passing from 14. (with pure strategies) to 17. (with mixed strategies).

6 The indirect argument, as outlined above, gives only necessary conditions. Hence

it may establish absurdity ( per absurdum proof), or narrow down the possibilities to one;

but in the latter case it is still necessary to show that the one remaining possibility is
satisfactory.

MIXED STRATEGIES. THE SOLUTION

149

17.4. The Minorant and the Majorant Game* (For Mixed Strategies)

17.4.1. Our present picture is then that player 1 chooses an arbitrary
— ■ ) ■' 1 ►
element £ from Sp and that player 2 chooses an arbitrary element 17 from

s fi .

Thus if player 1 wishes to play the strategy 2^ only, he will choose for

■■ 'E — ►

£ the coordinate vector 6 T i (cf. 16.1.3.) ; similarly for player 2, the strategy
and the vectors rj and 5 T *.

We imagine again that player 1 makes his choice of £ in ignorance of

player 2’s choice of 77 and vice versa.

The meaning is, of course, that when these choices have been made
player 1 will actually use (every) ti = 1, • * • , with the probabilities
£ Ti and the player 2 will use (every) r% = 1, • • • , 0* with the probabilities
i7 v Since their choices are independent, the mathematical expectation
of the outcome is

(17:2)

fix fit

K( £ , v ) = £ £ 3€( r i, tj)£ Ti ij v

r x -l r,-l

In other words, we have replaced the original game r by a new one of
essentially the same structure, but with the following formal difference*:
The numbers r h t 2 — the choices of the players — are replaced by the vectors
— ► — ►

£ , rj . The function 3 C(ti, t 2 ) — the outcome, or rather the “mathematical

expectation” of the outcome of a play — is replaced by K( £ , >7 ). All these
considerations demonstrate the identity of structure of our present view
of T with that of 14.1.2., — the sole difference being the replacement of

n, r 2 , 3 C(ti, r 2 ) by £ , rj , K( £ , rj ), described above. This isomorphism
suggests the application of the same devices which we used on the original
T, the comparison with the majorant and minorant games and r* as
described in 14.2., 14.3.1., 14.3.3.

17.4,2. Thus in r 1 player 1 chooses his £ first and player 2 chooses his 17

afterwards in full knowledge of the £ chosen by his opponent. In r 2 the
order of their choices is reversed. So the discussion of 14.3.1. applied

literally. Player 1, choosing a certain $ , may expect that player 2 will

choose his <j , so as to minimize K( £ , rj ) ; i.e. player l’s choice of £ leads to

the value Min - * K( £ , t ) ). This is a function of £ alone; hence player 1
— ► — > ►

should choose his £ so as to maximize Min-* K( £ , q ). Thus the value of
a play of r» is (for player 1)

160

ZERO-SUM TWO-PERSON GAMES: THEORY

vj = Max— Min— K( { , ij ).

£

Similarly the value of a play of r 2 (for player 1) turns out to be

v 2 = Min-+ Max-* K( £ , rj ).

(The apparent assumption of rational behavior of the opponent does not
really matter, since the justifications (14:A:a)-(14:A:e), (14:B:a)-(14:B:e)
of 14.3.1. and 14.3.3. again apply literally.)

As in 14.4.1. we can argue that the obvious fact that Ti is less favorable
for player 1 than r 2 constitutes a proof of

vi ^ v' 2 ,

and that if this is questioned, a rigorous proof is contained in (13 :A*) in

13.4.3. The x ) y , <f> there correspond to our £ , 7? , K. 1 If it should happen
that

v'i = vj,

then the considerations of 14.5. apply literally. The arguments (14:C:a)-
(14:C:f), (14:D:a), (14:D:b) loc. cit., determine the concept of a “ good ” £
and tj and fix the “value” of a play of (for the player 1) at

v' = v; = v' 2 . 2

All this happens by (13 :B*) in 13.4.3. if and only if a saddle point of K
exists. (The x ) y, <t> there correspond to our £ , rj , K.)

17.5. General Strict Determinateness

17 . 5 . 1 . We have replaced the Vi, v 2 of (14:A:c) and (14:B:c) by our
present vj, v' 2 , and the above discussion shows that the latter can perform
the functions of the former. But we are just as much dependent upon
v[ — v' 2 as we were then upon v x = v 2 . It is natural to ask, therefore,
whether there is any gain in this substitution.

Evidently this is the case if, as and when there is a better prospect
of having v( = v 2 (for any given r) than of having v x = v 2 . We called T
strictly determined when Vi = v 2 ; it now seems preferable to make a dis-
tinction and to designate r for Vi = v 2 as specially strictly determined , and
for v[ = v' 2 as generally strictly determined. This nomenclature is justified
only provided we can show that the former implies the latter.

1 Although £ , 17 are vectors, i.e. sequences of real numbers (£i, •••,£!, and

"i

171 , • • • , 17 ^) it is perfectly admissible to view each as a single variable in the maxima and
minima which we are now forming. Their domains are, of course, the sets Sfl , S$
which we introduced in 17.2.

* For an exhaustive repetition of the arguments in question cf. 17.8.

MIXED STRATEGIES. THE SOLUTION

151

This implication is plausible by common sense : Our introduction of mixed
strategies has increased the player's ability to defend himself against having
his strategy found out; so it may be expected that v' x , v 2 actually lie between
Vi, v 2 . For this reason one may even assert that

(17:3) Vi ^ v[ ^ v' 2 g v 2 .

(This inequality secures, of course, the implication just mentioned.)

To exclude all possibility of doubt we shall give a rigorous proof of
(17:3). It is convenient to prove this as a corollary of another lemma.
17 . 5 . 2 . First we prove this lemma:

(17 :A) For every { in

> } 01 02

Min-* K( £ , r, ) = Min-* V V 3C(r 1( r 2 )£ r nr,

T} 77 , ,

Ti-1 r 2 = l
01

= Min Tj £ 3C(ri, r 2 )£ v

T, = l

>

For every rj in Sp t

Max-* K( £ , r) ) = Max-* £ £ 3C(n, r 2 )f T ^ T!

S r,-lr,-l

= Max r[ 2 ) 3C(ti, Tl)T)r t .

t 2 - 1

Proof: We prove the first formula only; the proof of the second is exactly
the same, only interchanging Max and Min as well as ^ and

Consideration of the special vector rj = 5 T * (cf. 16.1.3. and the end of
17.2.) gives

Min-> X X ^( Tl > r *)irflr t ^ X X 3C( T 1> T2 ){ Ti 6 t / s = ^ X( Jl , r ' 2 ){ Tl .

r x *»l Tj*-1 T 1 1 T 2 ™ 1 r,®!

Since this is true for all r 2 , so

(17:4a) Min-* £ t 3C(n, ^ Min r ; £ 3C(n, ri)£ v

On the other hand, for all r 2

0i 0i

% t 2 )£,, ^ Min Tj 3C(r,,

r x -l r x -l

Given any r\ in Sp %) multiply this by r\ J% and sum over r 2 = 1, • • • , 0 2 .
a

Since £ iy T| = 1, therefore

r,-l

152

ZERO-SUM TWO-PERSON GAMES: THEORY

X X 3C(ri, T*)£ Ti ij, t ^ Min, t X 3C(ti, r t )( Tl

Ti- 1 r,-l T|-l

results. Since this is true for all rj , so

(17 :4 :b) Min-* £ X ^( r i> T »)fr l T ?r 1 ^ Min Tj £ 3C(ri, ri)£ v

11 r , - 1 r , - 1 n - 1

(17:4:a), (17 :4:b) yield together the desired relation.

If we combine the above formulae with the definition of v' lf v' 2 in 17.4.,
then we obtain

(17:5 :a)
(17:5:b)

v', = Max-* Min Tj X x ( T h T *)$v

{ r,-l

fi,

y't = Min-* Max. V X (n, r 2 )»/ T .

T,-l

These formulae have a simple verbal interpretation: In computing v' x we

need only to give player 1 the protection against having his strategy found

— >

out which lies in the use of £ (instead of ri) ; player 2 might as well proceed

in the old way and use r 2 (and not rj ). In computing v 2 the roles are inter-
changed. This is plausible by common-sense: v[ belongs to the game Ti
(cf. 17.4. and 14.2.); there player 2 chooses after player 1 and is fully
informed about the choice of player 1, — hence he needs no protection against
having his strategy found out by player 1. For v£ which belongs to the
game T* (cf. id.) the roles are interchanged.

Now the value of v[ becomes g if we restrict the variability of £ in

— ► — ► ,

the Max-^ of the above formula. Let us restrict it to the vectors £ = 6 r i

W - 1,

, 0i, cf. 16.1.3. and the end of 17.2.). Since

fix

X 3C(n, r s )« v ; = X (t[, Tt),

T,-l

this replaces our expression by

Max/ Min r# 3C(ri, t 2 ) = Vi.

So we have shown that

V! g v ;.

Similarly (cf . the remark at the beginning of the proof of our lemma above)
— > — > — ► ,

restriction of y to the y = S r « establishes

v» £ V'f

MIXED STRATEGIES. THE SOLUTION

153

Together with v[ ^ v 2 (cf. 17.4.), these inequalities prove
(17:3) vi ^ Vi ^ v' 2 g Vt,

as desired.

17.6. Proof of the Main Theorem

17.6. We have established that general strict determinateness (v' x = vj)
holds in all cases of special strict determinateness (vi = v 2 ) us is to be
expected. That it holds in some further cases as well — i.e. that we can
have Vj = v' 2 but not Vi = v 2 — is clear from our discussions of Matching
Pennies and Stone, Paper, Scissors. 1 Thus we may say, in the sense of
17.5.1. that the passage from special to general strict determinateness does
constitute an advance. But for all we know at this moment this advance
may not cover the entire ground which should be controlled; it could hap-
pen that certain games T are not even generally strictly determined, — i.e.
we have not yet excluded the possibility

v'i < v'.

If this possibility should occur, then all that was said in 14.7.1. would
apply again and to an increased extent : finding out one’s opponent’s strategy
would constitute a definite advantage

A' = v 2 — vi > 0,

and it would be difficult to see how a theory of the game should be con-
structed without some additional hypotheses as to “who finds out whose
strategy.”

The decisive fact is, therefore, that it can be shown that this never
happens. For all games T

v'i = v' 2
i.e.

(17:6) Max-* Min-» K( £ , rj ) = Min-> Max-* K( £ , t; ),

or equivalently (again use (13:B*) in 13.4.3. the x, y } <t> there corresponding
to our £ , rj , K) : A saddle point of K( £ , rj ) exists.

This is a general theorem valid for all functions K( £ , rj ) of the form

(17:2)

0i fit

K( ( , v ) = X Z x ( Tt > r *)Mv

Tl-l T,-l

The coefficients 3C(ri, r$) are absolutely unrestricted; they form, as described

— ► — >

in 14.1.3. a perfectly arbitrary matrix. The variables £ , i j are really

1 In both games Vi = — 1, v* ** 1 (cf. 14.7.2., 14.7.3.), while the discussion of 17.1.
can be interpreted as establishing vj = vj = 0.

154

ZERO-SUM TWO-PERSON GAMES: THEORY

sequences of real numbers £ 1 , • • • , £ #i and rj 1; ■ • • , ; their domains

— ► — >

being the sets Sp t (cf. footnote 1 on p. 150). The functions K( £ , rj )
of the form (17:2) are called bilinear forms.

With the help of the results of 16.4.3. the proof is easy . 1 This is it:

We apply (16:19:a), (16:19:b) in 16.4.3. replacing the i, j, n , m, a(i y j)

there by our r h t 2 , 0i, /3 2 , 3C(ti, r 2 ) and the vectors w , a: there by our £ , 77 .

If (16:19:b) holds, then we have a £ in Sp x with

2 3C(n, r s )f ri £ 0 for r 2 = 1,

ri-l

i.e. with

Min Tj 0C(ri, t 2 )£ Ti ^ 0.

ri-l

Therefore the formula (17:5 :a) of 17.5.2. gives

v' x ^ 0.

If (16:19:a) holds, then we have an rj in Sp 2 with

% 3C(ti, r 2 )rj Ti ^ 0 for

T,-l

n = 1 ,

Pu

1 This theorem occurred and was proved first in the original publication of one of the
authors on the theory of games: J. von Neumann: “Zur Theorie der Gesellschaftsspiele,”
Math. Annalen, Vol. 100 (1928), pp. 295-320.

A slightly more general form of this Min- Max problem arises in another question of
mathematical economics in connection with the equations of production :

J. von Neumann: “tJber ein okonomisches Gleichungssystem und eine Verall-
gemeinerung des Brouwer’schen Fixpunktsatzes,” Ergebnisse eines Math. Kolloquiums,
Vol. 8 (1937), pp. 73-83.

It seems worth remarking that two widely different problems related to mathe-
matical economics — although discussed by entirely different methods — lead to the same
mathematical problem, — and at that to one of a rather uncommon type: The “ Min-Max
type.” There may be some deeper formal connections here, as well as in some other
directions, mentioned in the second paper. The subject should be clarified further.

The proof of our theorem, given in the first paper, made a rather involved use of
some topology and of functional calculus. The second paper contained a different proof,
which was fully topological and connected the theorem with an important device of
that discipline: the so-called “ Fixed Point Theorem” of L. E. J. Brouwer. This aspect
was further clarified and the proof simplified by S. Kakutani: “A Generalization of
ttrouwer’s Fixed Point Theorem,” Duke Math. Journal, Vol. 8 (1941), pp. 457-459.

All these proofs are definitely non-elementary. The first elementary one was given
by J. Ville in the collection by E. Borel and collaborators, “Trait6 du Calcul des Prob-
ability et de ses Applications,” Vol. IV, 2: “Applications aux Jeux de Hasard,” Paris
(1938), Note by J. Ville : “Sur la Theorie G6n6rale des Jeux oti intervient FHabilet^ des
Joueura,” pp. 105-113.

The proof which we are going to give carries the elementarization initiated by
J. Ville further, and seems to be particularly simple. The key to the procedure is, of
course, the connection with the theory of convexity in 16. and particularly with the
results of 16.4.3.

MIXED STRATEGIES. THE SOLUTION

155

i.e. with

Max T| 3C (ri, t 2 )jj t> g 0.

r,- 1

Therefore the formula (17:5:b) of 17.5.2. gives

v' 2 £ 0.

So we see: Either ^ 0 or v 2 g 0, i.e.

(17:7) Never v', < 0 < v' 2 .

Now choose an arbitrary number w and replace the function 3 C(ti, t 2 )
by 3C(n, t 2 ) — w. 1

> * y } 0 1 02 > )

This replaces K( £ , y ) by K( £ , ) - w £ £ £r,*? v that is— as £ , y

t i - 1 r, = 1

01 02

lie in and so £ { Ti = ^ rj Ti — 1 — by K( £ , rj ) — w. Consequently

Tl -1 Tj =-1

v'i, v' 2 are replaced by v[ — w, v' 2 — w. 2 Therefore application of (17:7) to
these v'x — w, v 2 — w gives

(17:8)

Never v\ < w < v 2 .

Now w was perfectly arbitrary. Hence for v^ < v 2 it would be possible
to choose w with v[ < w < v 2 thus contradicting (17:8). So v[ < v' 2 is
impossible, and we have proved that v[ = v 2 as desired. This completes
the proof.

17.7. Comparison of the Treatments by Pure and by Mixed Strategies

17 . 7 . 1 . Before going further let us once more consider the meaning of
the result of

v'i = v' 2 .

The essential feature of this is that we have always v[ = v 2 but not always
Vi = v 2 , — i.e. always general strict determinateness, but not always special
strict determinateness (cf. the beginning of 17.6.).

Or, to express it mathematically:

We have always

(17:9) Max-* Min-* K( £ , rj ) — Min-* Max-* K( £ , rj ),

, $ v n Z

1 I.e. the game r is replaced by a new one which is played in precisely the same way as
T except that at the end player 1 gets less (and player 2 gets more) by the fixed amount w
than in T.

2 This is immediately clear if we remember the interpretation of the preceding
footnote.

156

ZERO-SUM TWO-PERSON GAMES: THEORY

h fix

(17:10) Max-* Min-* £ 3C(ri, =

’r.-lr.-l

fix fit

Min-* Max-* £ £ 3 C ( t i> T »)f* ,»?*,•

Using (17 :A) we may even write for this

*i />«

(17:11) Max-^*Min, t £ 3 C(ti, r a )£ ri = Min-^* Max r _ £ 3 C(ti, tj)tj v

ri - 1 r, - 1

But we do not always have

(17:12) Max Ti Min Tf 3C(ri, r 2 ) = Min Tj Max Ti 3C(ri, r 2 ).

Let us compare (17:9) and (17:12): (17:9) is always true and (17:12)

— ► — >

is not. Yet the difference between these is merely that of f, rj , K and n, r 2 ,
X. Why does the substitution of the former for the latter convert the
untrue assertion (17:12) into the true assertion (17:9)?

The reason is that theX(Ti, r 2 ) of (17:12) is a perfectly arbitrary func-
tion of its variables r h r 2 (cf. 14.1.3.), while the K( £ , rj ) of (17:9) is an

— > —¥

extremely special function of its variables £ , rj — i.e. of the £i, • • • , & t)
91 , • • * , — namely a bilinear form. (Cf. the first part of 17.6.) Thus
the absolute generality of 3 C(ti, t 2 ) renders any proof of (17:12) impossible,

— ► — y

while the special — bilinear form — nature of K( £ , rj ) provides the basis for
the proof of (17:9), as given in 17.6. 1

17.7.2, While this is plausible it may seem paradoxical that K( £ , vj )
should be more special than 3C(t i, r 2 ), although the former obtained from
the latter by a process which bore all the marks of a generalization: We
obtained it by the replacement of our original strict concept of a pure
strategy by the mixed strategies, as described in 17.2.; i.e. by the replace-
■— > — >

ment of n, r 2 by £ , 17 .

— > — >

But a closer inspection dispels this paradox. K( £ , ) is a very special

function when compared with 3C(r 1 , r 2 ) ; but its variables have an enormously

1 That the K( £ , 17 ) is a bilinear form is due to our use of the “ mathematical expecta-
tion” wherever probabilities intervene. It seems significant that the linearity of this
concept is connected with the existence of a solution, in the sense in which we found one.
Mathematically this opens up a rather interesting perspective: One might investigate
which other concepts, in place of “mathematical expectation,” would not interfere with
our solution, — i.e. with the result of 17.6. for zero-sum two-person games.

The concept of “mathematical expectation” is clearly a fundamental one in many
ways. Its significance from the point of view of the theory of utility was brought forth
particularly in 3.7.1.

MIXED STRATEGIES. THE SOLUTION

157

wider domain than the previous variables n, T 2 . Indeed r i had the finite

set (1, • • • , pi) for its domain, while £ varies over the set which is a

(Pi — l)-dimensional surface in the /Si-dimensional linear space (cf. the

— >

end of 16.2.2. and 17.2.). Similarly for r 2 and rj . l

There are actually among the £ in special points which correspond

to the various t\ in (1, • • • , Pi). Given such a ri we can form (as in

— > — >

16.1.3. and at the end of 17.2.) the coordinate vector £ = 8\ expressing

the choice of the strategy to the exclusion of all others. We can corre-
— ►

late special rj in with the r 2 in (1, • • • , p 2 ) in the same way : Given such

a r 2 we can form the coordinate vector rj = 8 r *, expressing the choice of the
strategy 2J* to the exclusion of all others.

Now clearly:

Pi /9*

K( 8 r i, 8 r i) = ^ X 3C(ti, r 2 ) 6r |r (

/ - 1 / - 1
1 1

= 3C(ri, r 2 ). 2

Thus the function K( £ , rj ) contains, in spite of its special character, the
entire function X(ri, t 2 ) and it is therefore really the more general concept
of the two, as it ought to be. It is actually more thanX(rj, r 2 ) since not all

( , rj are of the special form 5 r », 5 r 2 , — not all mixed strategies are pure.*

One could say that K( £ , rj ) is the extension of X(ri, t 2 ) from the narrower

— > — ► — > — ►

domain of theri,r 2 — i.e. of the 5 \ 8 r i — to the wider domain of the £, rj — i.e.

to all of Sp y} Sj3 t — from the pure strategies to the mixed strategies. The fact
— ► — ►

that K( £ , rj ) is a bilinear form expresses merely that this extension is carried
out by linear interpolation. That it is this process which must be used, is
of course due to the linear character of “ mathematical expectation.^ 4

— i

1 Observe that £ * I £i, • • • , £^l with the components £ Tl , r\ = 1 , • • ■ , 0 i, also

contains n ; but there is a fundamental difference. In 3 C(n, ri), n itself is a variable. In
— > — > — > — ►

K( ^ , rj ), £ is a variable, while n is, so to say, a variable within the variable. £ is
actually a function of ri (cf. the end of 16 . 1 . 2 .) and this function as such is the variable

of K( £ , ri ). Similarly for r 2 and 77 .

Or, in terms of n, r 2 : 3C(n, r 2 ) is a function of n, r 2 while K( £ , 77 ) is a function
of functions of n, * 2 (in the mathematical terminology: a functional).

2 The meaning of this formula is apparent if we consider what choice of strategies

8 r i, 8 T t represent.

3 I.e. several strategies may be used effectively with positive probabilities.

4 The fundamental connection between the concept of numerical utility and the linear
“ mathematical expectation ” was pointed out at the end of 3 . 7 . 1 .

158

ZERO-SUM TWO-PERSON GAMES: THEORY

17 . 7 . 3 . Reverting to (17:9)-(17:12), we see now that we can express the
truth of (17 :9)-(17-l 1) and the untruth of (17:12) as follows:

(17:9), (17:10) express that each player is fully protected against having
his strategy found out by his opponent if he can use the mixed strategies

£ , ri instead of the pure strategies r h r 2 . (17 :11) states that this remains

true if the player who finds out his opponent’s strategy uses the r h r 2

while only the player whose strategy is being found out enjoys the pro-
— > — >

tectionofthe £ , rj . Thefalsityof (17:12), finally, shows that both players —
and particularly the player whose strategy happens to be found out — may

— ► — ►

not forego with impunity the protection of the £ , tj .

17.8. Analysis of General Strict Determinateness

17 . 8 . 1 . We shall now reformulate the contents of 14.5. — as mentioned
at the end of 17.4. — with particular consideration of the fact established
in 17.6. that every zero-sum two-person game r is generally strictly deter-
mined. Owing to this result we may define :

v' = Max— Min— K( $ , rj ) = Min— Max— K( £ , rj )

in n i

= Sa-|-K( £ , rj )•

i I v

(Cf. also (13 :C*) in 13.5.2. and the end of 13.4.3.)

Let us form two sets A, B — subsets of Sp, Sfi t , respectively — in analogy
to the definition of the sets A, B in (14:D:a), (14:D:b) of 14.5.1. These
are the sets A+, B+ of 13.5.1. (the </> corresponding to our K). We define:

(17:B:a) A is the set of those £ (in Sp ) for which Min— K( £ , rj )

1 n

assumes its maximum value, i.e. for which

Min- K( £ , v ) = Max- Min- K( £ , r , ) = v'.

n in

(17:B:b) B is the set of those rj (in Sp t ) for which Max-^ K( £ , rj )

assumes its minimum value, i.e. for which

Max— K( £ , r] ) — Min— Max— K( £ , rj ) = v'.

i *i

It is now possible to repeat the argumentation of 14.5.

In doing this we shall use the homologous enumeration for the assertions
(14:C:a)-(14:C:f) as in 14.5. 1

1 (a)-(f) will therefore appear in an order different from the natural one. This was
equally true in 14.5., since the enumeration there was based upon that of 14.3.1., 14.3.3.,
and the argumentation in those paragraphs followed a somewhat different route.

MIXED STRATEGIES. THE SOLUTION

159

We observe first:

(17:C:d) Player 1 can, by playing appropriately, secure for himself

a gain ^ v', — irrespective of what player 2 does.

Player 2 can, by playing appropriately, secure for himself
a gain ^ — v', — irrespective of what player 1 does.

Proof : Let player 1 choose £ from A ; then irrespective of what player 2
does, i.e. for every rj we have K( £ , rj ) ^ Min— K( £ , n ) = v'. Let player

V

2 choose rj from B. Then irrespective of what player 1 does, i.e. for every
£ , we have K( £ , rj ) g Max— K( £ , rj ) = v\ This completes the proof.
Second, (17:C:d) is clearly equivalent to this:

(17:C:e) Player 2 can, by playing appropriately, make it sure that

the gain of player 1 is ^ v', i.e. prevent him from gaining

> v', irrespective of what player 1 does.

Player 1 can, by playing appropriately, make it sure that
the gain of player 2 is ^ — v', i.e. prevent him from gaining

> — v', irrespective of what player 2 does.

17 . 8 . 2 . Third, we may now assert — on the basis of (17:C:d) and (17:C:e)
and of the considerations in the proof of (17:C:d) — that:

(17:C:a) The good way (combination of strategies) for 1 to play

— > _ _

the game r is to choose any £ belonging to A , — A being the set
of (17:B:a) above.

(17:C:b) The good way (combination of strategies) for 2 to play

the game T is to choose any tj belonging to JS, — B being the set
of (17:B:b) above.

Fourth, combination of the assertions of (17:C:d) — or equally well
of those of (17:C:e) — gives:

(17:C:c) If both players 1 and 2 play the game T well — i.e. if £

belongs to A and rj belongs to B — then the value of K( £ , rj )
will be equal to the value of a play (for 1), — i.e. to v\

We add the observation that (13:D*) in 13.5.2. and the remark concerning
the sets A, B before (17:B:a), (17:B:b) above give together this:

Both players 1 and 2 play the game T well — i.e. £ belongs

_ — ► — > — >

to A and ij belongs to B — if and only if £ , rj is a saddle point

(17:C:f)

160

ZERO-SUM TWO-PERSON GAMES: THEORY

All this should make it amply clear that v' may indeed be interpreted
as the value of a play of T (for 1), and that A, 5 contain the good ways of
playing T for 1, 2, respectively. There is nothing heuristic or uncertain
about the entire argumentation (17:C:a)-(17:C:f). We have made no
extra hypotheses about the “intelligence” of the players, about “who
has found out whose strategy” etc. Nor are our results for one player
based upon any belief in the rational conduct of the other, — a point the
importance of which we have repeatedly stressed. (Cf. the end of 4.1.2.;
also 15.8.3.)

17.9. Further Characteristics of Good Strategies

17.9.1. The last results — (17:C:c) and (17 :C :f) in 17.8.2. — give also a
simple explicit characterization of the elements of our present solution, —
i.e. of the number v' and of the vector sets A and B .

By (17:C:c) loc. cit., A, B determine v'; hence we need only study A,
5, and we shall do this by means of (17 :C :f) id.

— > _ — > _

According to that criterion, $ belongs to A, and tj belongs to B if and

only if £ , 77 is a saddle point of K( £ , rj ). This means that

K( £ , v ) =

Max- K( £ „)

{ Min- K( £ , v ')

V

We make this explicit by using the expression (17:2) of 17.4.1. and
17.6. for K( £ , rj ), and the expressions of the lemma (17:A) of 17.5.2. for
Max—, K( { rj) and Min- K( £ , rj ')• Then our equations become:

01 01

X I) 3 C( t »> T i)^r l Vr, =

Tf - 1 r, - 1

0 ,

Max,' £

T,-l

0 X

Min T ' X r 2 )£*,

r,-l

0 i 0 ,

Considering that ^ £ Ti = ^ rj Tt = 1 , we can also write for these
r » - 1 - 1

X [Max,; | £ 3C(rl, - £ K ( r b rs)»jr t ] = 0,

rj-1 r,-l r,-l

0f 0| 0f

% [-Min r ; { £ x ( T u T s)^,} + % W( T u T *)fr,] »?., = 0.

r, ■» 1 r» 1 r, « 1

Now on the left-hand side of these equations the £ v *?r f have coefficients
which are all ^ 0 . 1 The £ Ti , »?r f themselves are also ^ 0. Hence these
1 Observe how the Max and Min occur there !

MIXED STRATEGIES. THE SOLUTION

161

equations hold only when all terms of their left hand sides vanish separately.
I.e. when for each n = 1,. • * * , /Si for which the coefficient is not zero, we
have = 0; and for each 72 = 1, * • • , fa for which the coefficient is not
zero, we have r ) Tf = 0.

Summing up;

(17 :D) £ belongs to A and rj belongs to B if and only if these

are true;

fit

For each n = 1, • • • , f) h for which £ 3C(ti, T 2 )rj Tt does

r,-l

not assume its maximum (in r i) we have £ Tl = 0.

fix

For each n = 1, • • • , 0*, for which £ 3C(ri, 1-2)$^ does

r,-l

not assume its minimum (in r 2 ) we have rj Ti = 0.

It is easy to formulate these principles verbally. They express this: If

£ , rj are good mixed strategies, then £ excludes all strategies n which are

— ► — ►

not optimal (for player 1) against rj , and 77 excludes all strategies r 2 which

are not optimal (for player 2) against £ ; i.e. £ , 77 are — as was to be
expected — optimal against each other.

17 . 9 . 2 . Another remark which may be made at this point is this:

(17:E) The game is specially strictly determined if and only if there

exists for each player a good strategy which is a pure strategy.

In view of our past discussions, and particularly of the process of gen-
eralization by which we passed from pure strategies to mixed strategies,
this assertion may be intuitively convincing. But we shall also supply a
mathematical proof, which is equally simple. This is it :

We saw in the last part of 17.5.2. that both Vi and v' x obtain by applying

0i ►

Max-> to Min Tj ^ X (n, t 2 )£ Ti , only with different domains for £ : The set
* * r i ™ 1

of all 5 h (ri = 1, • • • , 0i) for Vi, and all of Sp x for v' x ; i.e. the pure strategies
in the first case, and the mixed ones in the second. Hence Vi = v\, i.e.
the two maxima are equal if and only if the maximum of the second domain
is assumed (at least once) within the first domain. This means by (17 :D)
above that (at least) one pure strategy must belong to A, i.e. be a good one.
I.e.

Vi = vi if and only if there exists for player 1 a good
strategy which is a pure strategy.

(17:F:a)

162

ZERO-SUM TWO-PERSON GAMES: THEORY

Similarly:

(17:F:b) v 2 = v 2 if and only if there exists for player 2 a good

strategy which is a pure strategy.

Now v[ = v' 2 = v' and strict determinateness means Vi = v 2 = v',
i.e. Vi = vi and v 2 = v 2 . So (17:F:a), (17:F:b) give together (17:E).

17.10. Mistakes and Their Consequences. Permanent Optimality

17 . 10 . 1 . Our past discussions have made clear what a good mixed
strategy is. Let us now say a few words about the other mixed strategies.
We want to express the distance from “goodness” for those strategies (i.e.

vectors £ , tj) which are not good ; and to obtain some picture of the conse-
quences of a mistake — i.e. of the use of a strategy which is not good. How-
ever, we shall not attempt to exhaust this subject, which has many intriguing
ramifications.

— y — >

For any £ in 5^ and any ^ in Sg t we form the numerical functions

«( £ ) = v' — Min-* K( { , „),

(17:13:a)

(17:13:b)

By the lemma (17:A) of 17.5.2. equally

(17:13:a*)

(17:13:b*)

The definition

£( v ) = Max- K( £ , »? ) - v'.

»s

«( £) = v' — Min ri £ 3C(n, r 2 )£ ri ,

Ti-1

/3( v) = Max f| V 3C(ri, T t )vr t - v'.

r,-l

v' = Max— Min- K( f , v) = Min- Max- K( £ , jj )

t V V t

guarantees that always

a( £ ) k 0, 0( v) ^ 0.

And now (17:B:a), (17:B:b) and (17:C:a), (17:C:b) in 17.8. imply that {

— y — y — y

is good if and only if a( £ ) =0, and ij is good if and only if 0( 17 ) =0.

— y — y — y — ►

Thus a( £ ), /3( r ) ) are convenient numerical measures for the general £ , 77
expressing their distance from goodness. The explicit verbal formulation

of what a( £ ), 0( 1? ) are, makes this interpretation even more plausible: The
formulae (17:13:a), (17:13:b) or (17:13:a*), (17:13:b*) above make clear

MIXED STRATEGIES. THE SOLUTION

163

how much of a loss the player risks — relative to the value of a play for him 1
— by using this particular strategy. We mean here “risk” in the sense
of the worst that can happen under the given conditions . 2 3 * * * *

It must be understood, however, that a (£),£(*?) do not disclose which
strategy of the opponent will inflict this (maximum) loss upon the player who

is using {or 77 . It is, in particular, not at all certain that if the opponent

uses some particular good strategy, i.e. an rj 0 in 5 or a { 0 in A f this in itself

— ► — ►

implies the maximum loss in question. If a (not good) { or ^ is used by the

— ► — y

player, then the maximum loss will occur for those »j ' or { ' of the opponent,
for which

(17:14:a)

(17:14:b)

K( € , v') = Min— K( { , „ ),

V

K(7,7) = Max- K(1,7),

i.e. if rj ' is optimal against the given { , or { ' optimal against the given rj .

— y — >

And we have never ascertained whether any fixed rj 0 or £ 0 can be optimal
against all { or 77 .

— ► — ►

17.10.2. Let us therefore call an ij' ora £ ' which is optimal against all

£ or 1 ; — i.e. which fulfills (17:14:a), or (17:14:b) in 17.10.1. for all £, »; —

► — ►

permanently optimal. Any permanently optimal rj ' or { ' is necessarily
good; this should be clear conceptually and an exact proof is easy . 8 But

1 I.e. we mean by loss the value of the play minus the actual outcome: v' — K( £ , 1 7 )

for player 1 and ( — v') — ( — K( £ , 17 )) - K( £ , 17 ) — v' for player 2.

* 2 Indeed, using the previous footnote and (17:13:a), (17 :13:b)

«( 7 ) = v' - Min-* K(7. 7) = Max-* {v' - K(7. 7)1,

0 ( 7 ) = Max- K(7, 7) - V' - Max- (K(7, 7) - v').

I.e. each is a maximum loss.

— y — >

3 Proof : It suffices to show this for 77 the proof for £ ' is analogous.

Let 77 'be permanently optimal. Choose a £ * which is optimal against 17 ', i.e. with

By definition

K(« *, *') = Max— K( £ , „ ')

K(«*, n') = Min— K( { *, 77).

Thus £ *, 17 ' is a saddle point of K( £ , 77 ) and therefore 77 ' belongs to £ — i.e. it is good-

by (17:C:f) in 17.8.2.

164

ZERO-SUM TWO-PERSON GAMES: THEORY

the question remains: Are all good strategies also permanently optimal?
And even: Do any permanently optimal strategies exist?

In general the answer is no. Thus in Matching Pennies or in Stone,
Paper, Scissors, the only good strategy (for player 1 as well as for player 2) is

$ = v = {i, i} or {i, i, i) , respectively. 1 If player 1 played differently —
e.g. always “heads” 2 or always “stone” 2 — then he would lose if the oppo-
nent countered by playing “tails” 3 or “paper.” 3 But then the opponent's
strategy is not good — i.e. {-£, -£} or {£, i, i}, respectively — either. If the
opponent played the good strategy, then the player's mistake would not
matter. 4

We shall get another example of this — in a more subtle and complicated
way — in connection with Poker and the necessity of “bluffing,” in 19.2 and

19.10.3.

All this may be summed up by saying that while our good strategies
are perfect from the defensive point of view, they will (in general) not get
the maximum out of the opponent's (possible) mistakes, — i.e. they are not
calculated for the offensive.

It should be remembered, however, that our deductions of 17.8. are
nevertheless cogent; i.e. a theory of the offensive, in this sense, is not
possible without essentially new ideas. The reader who is reluctant to
accept this, ought to visualize the situation in Matching Pennies or in
Stone, Paper, Scissors once more; the extreme simplicity of these two
games makes the decisive points particularly clear.

Another caveat against overemphasizing this point is: A great deal
goes, in common parlance, under the name of “offensive,” which is not at
all “offensive” in the above sense, — i.e. which is fully covered by our pres-
ent theory. This holds for all games in which perfect information prevails,
as will be seen in 17. 10.3. 6 Also such typically “aggressive” operations (and
which are necessitated by imperfect information) as “bluffing” in Poker. 6

17 . 10 . 3 . We conclude by remarking that there is an important class of
(zero-sum two-person) games in which permanently optimal strategies
exist. These are the games in which perfect information prevails, which
we analyzed in 15. and particularly in 15.3.2., 15.6., 15.7. Indeed, a small
modification of the proof of special strict determinateness of these games, as
given loc. cit., would suffice to establish this assertion too. It would give
permanently optimal pure strategies. But we do not enter upon these con-
siderations here.

1 Cf. 17.1. Any other probabilities would lead to losses when “found out. ” Cf. below.

* This is $ = $ ' = 1 1, 0) or { 1, 0, 0) , respectively.

* This is i? - *" - {0, 1| or (0, 1, 0), respectively.

4 I.e. the bad strategy of “heads” (or “stone”) can be defeated only by “tails” (or
“paper”), which is just as bad in itself.

1 Thus Chess and Backgammon are included.

•The preceding discussion applies rather to the failure to “bluff.” Cf. 19.2. and

19 . 10 . 3 .

MIXED STRATEGIES. THE SOLUTION

165

Since the games in which perfect information prevails are always spe-
cially strictly determined (cf. above), one may suspect a more fundamental
connection between specially strictly determined games and those in which
permanently optimal strategies exist (for both players). We do not intend
to discuss these things here any further, but mention the following facts
which are relevant in this connection :

(17 :G :a) It can be shown that if permanently optimal strategies exist

(for both players) then the game must be specially strictly
determined.

(17:G:b) It can be shown that the converse of (17:G:a) is not true.

(17:G:c) Certain refinements of the concept of special strict deter-

minateness seem to bear a closer relationship to the existence
of permanently optimal strategies.

17.11. The Interchange of Players. Symmetry

17 . 11 . 1 . Let us consider the role of symmetry, or more generally the
effects of interchanging the players 1 and 2 in the game T. This will
naturally be a continuation of the analysis of 14.6.

As was pointed out there, this interchange of the players replaces the
function 3C(ri, r 2 ) by — 3 C(t 2 , ti). The formula (17:2) of 17.4.1. and 17. G.

— > —> — > — >

shows that the effect of this for K( £ , rj ) is to replace it by — K( r; , £ ). In
the terminology of 16.4.2., we replace the matrix (of 3C(r i, r 2 ) cf. 14.1.3.) by
its negative transposed matrix.

Thus the perfect analogy of the considerations in 14. continues; again
we have the same formal results as there, provided that we replace

r i, t 2 , 3C(ri, t 2 ) by £ , r) , K( £ , rj ). (Cf. the previous occurrence of this
in 17.4. and 17.8.)

We saw in 14.6. that this replacement of 3C(ri, t 2 ) by — 3 C(t 2 , n) carries
Vi, v 2 into — v 2 , — Vi. A literal repetition of those considerations shows

now that the corresponding replacement of K( f , rj) by — K( 77 , £)
carries vi, v 2 into — v 2 , — vi. Summing up: Interchanging the players 1, 2,
carries Vi, v 2 , v' 1? v' 2 into — v 2 , — Vi, — v' 2 , — v' t .

The result of 14.6. established for (special) strict determinateness was
that v = Vi = v 2 is carried into — v = — vi = —v 2 . In the absence of
that property no such refinement of the assertion was possible.

At the present we know that we always have general strict deter-
minateness, so that v' = v' x = v' 2 . Consequently this is carried into
— v' = — v'i = — v' 2 .

Verbally the content of this result is clear: Since we succeeded in defining
a satisfactory concept of the value of a play of T (for the player 1), v', it is
only reasonable that this quantity should change its sign when the roles
of the players are interchanged.

17 . 11 . 2 . We can also state rigorously when the game V is symmetric.
This is the case when the two players 1 and 2 have precisely the same

166

ZERO SUM TWO-PERSON GAMES: THEORY

role in it, — i.e. if the game T is identical with that game which obtains
from it by interchanging the two players 1, 2. According to what was said
above, this means that

or equivalently that

3C(ri, t 2 ) = — 3C(t 2 , ti),

K( { , v) = K( rj , {)•

This property of the matrix 3C(ri, t 2 ) or of the bilinear form K( { , tj ) was
introduced in 16.4.4. and called skew-symmetry , 1,2

In this case Vi, v 2 must coincide with — v 2 , — Vi; hence Vi = — v 2 ,
and since Vi ^ v 2 , so Vi ^ 0. But v' must coincid'e with — v'; therefore
we can even assert that

v' = 0. 1 * 3

So we see: The value of each play of a symmetrical game is zero.

It should be noted that the value v' of each play of a game T could be
zero without T being symmetric. A game in which v' = 0 will be called
fair.

The examples of 14.7.2., 14.7.3. illustrate this: Stone, Paper, Scissors is
symmetric (and hence fair); Matching Pennies is fair (cf. 17.1.) without
being symmetric. 4

1 For a matrix 3C(n, r 2 ) or for the corresponding bilinear form K( £ , tj ) symmetry
is defined by

3C(ri, r 2 ) = 3C(t 2 , ti),

or equivalently by

k(7,7) = k<7, 7).

It is remarkable that symmetry of the game V is equivalent to skew-symmetry, and not to
symmetry, of its matrix or bilinear form.

* Thus skew-symmetry means that a reflection of the matrix scheme of Fig. 15 in
14.1.3. on its main diagonal (consisting of the fields (1, 1), (2, 2), etc.) carries it into its
own negative. (Symmetry, in the sense of the preceding footnote, would mean that it
carries it into itself.)

Now the matrix scheme of Fig. 15 is rectangular; it has £ 2 columns and /3i rows. In
the case under consideration its shape must be unaltered by this reflection. Hence it
must be quadratic, — i.e. /3i = 0 2 . This is so, however, automatically, since the players
1, 2 are assumed to have the same role in r.

3 This is, of course, due to our knowing that v[ = \ 2 . Without this — i.e. without
the general theorem (16:F) of 16.4.3. — we should assert for the vj, v' 2 only the same which
we obtained above for the Vi, v 2 : vj = —vj and since vj ^ Vj, sov, ^ 0.

4 The players 1 and 2 have different roles in Matching Pennies: 1 tries to match,
and 2 tries to avoid matching. Of course, one has a feeling that this difference is inessen-
tial and that the fairness of Matching Pennies is due to this inessentiality of the assyme-
try. This could be elaborated upon, but we do not wish to do this on this occasion. A
better example of fairness without symmetry would be given by a game which is grossly
unsymmetric, but in which the advantages and disadvantages of each player are so
judiciously adjusted that a fair game — i.e. value v' = 0 — results.

A not altogether successful attempt at such a game is the ordinary way of “Rolling
Dice.” In this game player 1 — the “player" — rolls two dice, each of which bear the
numbers 1, • • • , 6. Thus at each roll any total 2, • • • , 12 may result. These totals

MIXED STRATEGIES. THE SOLUTION 167

In a symmetrical game the sets A, B of (17:B:a), (17:B:b) in 17.8. are

_ — > — >

obviously identical. Since A = B we may put £ = r\ in the final criterion
(17 :D) of 17.9. We restate it for this case:

(17:H) In a symmetrical game, £ belongs to A if and only if this is

true: For each r 2 = 1, * • * 0% for which 3C(ri, t 2 )£ Tj does

T,-l

not assume its minimum (in r 2 ) we have £ Tj = 0.

Using the terminology of the concluding remark of 17.9., we see that the

above condition expresses this : £ is optimal against itself.

17 . 11 . 3 . The results of 17.11.1., 17.11.2. — that in every symmetrical
game v' = 0 — can be combined with (17:C:d) in 17.8. Then we obtain
this:

(17:1) In a symmetrical game each player can, by playing appropri-

ately, avoid loss 1 irrespective of what the opponent does.

We can state this mathematically as follows:

If the matrix 3C(ri, t 2 ) is skew-symmetric, then there exists a vector £
in S Pi with

0i

X 3C(n, r,)* Ti £ 0 for r 2 = 1, • • • , 0*.

* 1*1

This could also have been obtained directly, because it coincides with
the last result (16 :G) in 16.4.4. To see this it suffices to introduce there
our present notations: Replace the i, j, a(i , j) there by our n, r 2 , 3€(ri, r 2 )

and the w there by our £ .

have the following probabilities:

Total

2

3

4

5

6

7

8

9

10

11

12

Chance out of 36

1

2

3

4

5

6

5

4

3

2

1

Probability

3*8

2

56

A

3 4 8

3*8

3 S 8

4

56

3

38

2

38

3*8

The rule is that if the “player” rolls 7 or 11 (“natural”) he wins. If he rolls 2, 3, or 12
he loses. If he rolls anything else (4, 5, 6, or 8, 9, 10) then he rolls again until he rolls
either a repeat of the original one (in which case he wins), or a 7 (in which case he loses).
Player 2 (the “house”) has no influence on the play.

In spite of the great differences of the rules as they affect players 1 and 2 (the
“player” and the “house”) their chances are nearly equal: A simple computation, which
we do not detail, shows that the “player” has 244 chances against 251 for the “house,”
out of a total of 495; i.e. the value of a play — played for a unit stake — is

244 - 251
495

_7_

495

-1.414%.

Thus the approximation to fairness is reasonably good, and it may be questioned whether
more was intended.

1 I.e. secure himself a gain ^ 0.

168

ZERO-SUM TWO-PERSON GAMES: THEORY

It is even possible to base our entire theory on this fact, i.e. to derive
the theorem of 17.6. from the above result. In other words: The general
strict determinateness of all r can be derived from that one of the symmetric
ones. The proof has a certain interest of its own, but we shall not discuss
it here since the derivation of 17.6. is more direct.

The possibility of protecting oneself against loss (in a symmetric game)

exists only due to our use of the mixed strategies {, rj (cf. the end of 17.7.).
If the players are restricted to pure strategies r h r 2 then the danger of having
one’s strategy found out, and consequently of sustaining losses, exists.
To see this it suffices to recall what we found concerning Stone, Paper,
Scissors (cf. 14.7. and 17.1.1.). We shall recognize the same fact in con-
nection with Poker and the necessity of “bluffing” in 19.2.1.

CHAPTER IV

ZERO-SUM TWO-PERSON GAMES: EXAMPLES

18. Some Elementary Games

18.1. The Simplest Games

18.1.1. We have concluded our general discussion of the zero-sum two-
person game. We shall now proceed to examine specific examples of such
games. These examples will exhibit better than any general abstract
discussions could, the true significance of the various components of our
theory. They will show, in particular, how some formal steps which are
dictated by our theory permit a direct common-sense interpretation. It
will appear that we have here a rigorous formalization of the main aspects
of such “practical” and “psychological” phenomena as those to be men-
tioned in 19.2., 19.10. and 19.16. 1

18.1.2. The size of the numbers 0i, — i.e. the number of alternatives

confronting the two players in the normalized form of the game — gives a
reasonable first estimate for the degree of complication of the game T.
The case that either, or both, of these numbers is 1 may be disregarded: This
would mean that the player in question has no choice at all by which he can
influence the game. 2 Therefore the simplest games of the class which
interests us are those with

(18:1) /Si = 02 = 2.

We saw in 14.7. that Matching Pennies is such a game; its matrix scheme
was given in Figure 12 in 13.4.1. Another instance of such a game is Figure
14, id.

1

2

1

3C(1, 1)

3C(1,2)

2

3C(2, 1)

3C(2, 2)

Figure 27.

Let us now consider the most general game falling under (18:1), i.e.
under Figure 27. This applies, e.g., to Matching Pennies if the various ways
of matching do not necessarily represent the same gain (or a gain at all),

1 We stress this because of the widely held opinion that these things are congenitally
unfit for rigorous (mathematical) treatment.

* Thus the game would really be one of one person ; but then, of course, no longer of
zero sum. Cf. 12.2.

169

170

ZERO-SUM TWO-PERSON GAMES: EXAMPLES

nor the various ways of not matching the same loss (or a loss at all). 1 We
propose to discuss for this case the results of 17.8., — the value of the game T
and the sets of good strategies A, 5 . These concepts have been established
by the general existential proof of 17.8. (based on the theorem of 17.6.);
but we wish to obtain them again by explicit computation in this special
case, and thereby gain some further insight into their functioning and their
possibilities.

18 . 1 . 3 . There are certain trivial adjustments which can be made on
a game given by Figure 27, and which simplify an exhaustive discussion
considerably.

First it is quite arbitrary which of the two choices of player 1 we denote
by ti = 1 and by ri = 2; we may interchange these, — i.e. the two rows of
the matrix.

Second, it is equally arbitrary which of the two choices of player 2 we
denote by r 2 = 1 and by r 2 = 2; we may interchange these, — i.e. the two
columns of the matrix.

Finally, it is also arbitrary which of the two players we call 1 and which
2; we may interchange these, — i.e. replace 3C(ri, t 2 ) by — X(r i, r 2 ) (cf. 14.6.
and 17.11.). This amounts to interchanging the rows and the columns
of the matrix, and changing the sign of its elements besides.

Putting everything together, we have here 2 X 2 X 2 = 8 possible
adjustments, all of which describe essentially the same game.

18.2. Detailed Quantitative Discussion of These Games

18 . 2 . 1 . We proceed now to the discussion proper. This will consist
in the consideration of several alternative possibilities, the “Cases” to be
enumerated below.

These Cases are distinguished by the various possibilities which exist
for the positions of those fields of the matrix where 3C(ri, t 2 ) assumes its
maximum and its minimum for both variables n, r 2 together. Their
delimitations may first appear to be arbitrary; but the fact that they
lead to a quick cataloguing of all possibilities justifies them ex post.

Consider accordingly Max r , Tj 3C(ri, t 2 ) and Min VTj X(r i, r 2 ). Each
one of these values will be assumed at least once and might be assumed more
than once; 2 but this does not concern us at this juncture. We begin now
with the definition of the various Cases :

18 . 2 . 2 . Case (A): It is possible to choose a field where Max TiTj is
assumed and one where Min T lTj is assumed, so that the two are neither in
the same row nor in the same column.

By interchanging r i = 1, 2 as well as r 2 = 1, 2 we can make the first-
mentioned field (of Max TilTj ) to be (1, 1). The second-mentioned field

1 Comparison of Figs. 12 and 27 shows that in Matching Pennies 3C(1, 1) = 3C(2, 2) = 1
(gain on matching); 3C(1, 2) = 3C(2, 1) = —1 (loss on not matching).

2 In Matching Pennies (cf. footnote 1 above) the Max^.r, is 1 and is assumed at
(1, 1) and (2, 2), while the Min rj , rj is —1 and is assumed at (1, 2) and (2, 1).

SOME ELEMENTARY GAMES

171

(of Min r . 7j ) must then be (2, 2). Consequently we have

(18:2) ^‘•»{iK(2;?)i| JC<2 ' 2) '

Therefore (1, 2) is a saddle point. 1

Thus the game is strictly determined in this case and

(18:3) v' = v = 30(1, 2).

18 . 2 . 3 . Case ( B ): It is impossible to make the choices as prescribed
above:

Choose the two fields in question (of Max T , ri and Min Ti , Tj ); then they
are in the same row or in the same column. If the former should be the case,
then interchange the players 1, 2, so that these two fields are at any rate
in the same column. 2

By interchanging ti = 1, 2 as well as r 2 = 1, 2 if necessary, we can again
make the first-mentioned field (of Max T tTi ) to be (1, 1). So the column in
question is r 2 = 1. The second-mentioned field (of Min T (T> ) must then be
(2, l). 3 Consequently we have:

( 18 : 4 ) « i - i )(iS2)i|*»- i >-

Actually 30(1, 1) = 30(1, 2) or 30(2, 2) = 30(2, 1) are excluded because
for the Max r , T) and Min r ,, t fields they would permit the alternative choices
of (1, 2), (2, 1) or (1, 1), (2, 2), thus bringing about Case (A). 4
So we can strengthen (18:4) to

(18:5) *M)|^S:2)>|«1)’

We must now make a further disjunction:

18 . 2 . 4 . Case (Bi):

(18:6) 30(1, 2) £ 30(2, 2)

Then (18:5) can be strengthened to

(18:7) 30(1, 1) > 30(1, 2) ^ 30(2, 2) > 30(2, 1).

Therefore (1, 2) is again a saddle point.

Thus the game is strictly determined in this case too; and again

(18:8) v' = v =30(1, 2).

1 Recall 13.4.2. Observe that we had to take (1,2) and not (2, 1).

2 This interchange of the two players changes the sign of every matrix element (cf.
above), hence it interchanges Max Tl ,r 2 and Min Ti , Tj . But they will nevertheless be in
the same column.

3 To be precise: It might also be (1, 1). But then 3C(n, r,) has the same Max Tl ,T s and
Min ri . Tj , and so it is a constant. Then we can use (2, 1) also for Min ri>Tj .

<3C(1, 1) - 30(2, 2) and 3C(1, 2) * 30(2, 1) are perfectly possible, as the example of
Matching Pennies shows. Cf. footnote 1 on p. 170 and footnote 1 on p. 172.

172

ZERO-SUM TWO-PERSON GAMES: EXAMPLES

18 . 2 . 5 . Case (B t ):

(18:9) 3C(1, 2) < JC(2, 2)

Then (18:5) can be strengthened to

(18:10) 3C(1, 1) ^ 3C(2, 2) > 3C(l, 2) £ 3C(2, l). 1

The game is not strictly determined. 2

It is easy however, to find good strategies, i.e. a £ in A and an jj in B, by
satisfying the characteristic condition (17 :D) of 17.9. We can do even more :

We can choose i) so that 2, 3C(ri, r*)ij T> is the same for all t i and f so that

T,“l

2

2) 3C(ti, t*)^ is the same for all tj. For this purpose we need:

T,-l

{ 3C(1, 1)tji + 3C(1, 2)ij» = X(2, l)iji + 3C(2, 2)r,s.
(3C(1, 1)?! +3C(2, 1){ 8 =3C(1, 2){, +3C(2, 2)fc.

(18:11)

This means
(18:12)

We must satisfy these ratios, subject to the permanent requirements

*i:b =3C(2, 2) -X(2, 1):0C(1, 1) - 3C(1, 2),
Vl : V2 =3C(2, 2) — 3C(1, 2):3C(1, 1) - 3C(2, 1).

£i ^ 0, $2 = 0 £i + f* — 1

Vl ^ 0, 1/2^0 T/l + J/2 = 1

This is possible because the prescribed ratios (i.e. the right-hand sides in
(18:12)) are positive by (18:10). We have

> = X(2, 2) - 3C(2, 1)

3C(1, 1) +3C(2, 2) -3C(1, 2) -3C(2, 1)’

> ac(l, 1) — 3C(1, 2)

3C(1, 1) + 3C(2, 2) - 3C(1, 2) - X(2, 1)

and further

ae(2, 2) - ac(i, 2)

,1 3C(1, 1) + 3C(2, 2) - 3C(1, 2) - 0C(2, 1)’

3C(1, 1) - 3C(2, 1)

Vt 3C(1, 1) + 3C(2, 2) -3C(1, 2) -3C(2, 1)'

We can even show that these £ , rj are unique, i.e. that -4, B possess no other
elements.

1 This is actually the case for Matching Pennies. Cf. footnotes l on p. 170 and 4
on p. 171.

* Clearly Vi * Max Ti Min fl 3C(n, tj) - 3C(1, 2), vj - Min r# Max Tl 3C(n, r%) - JC(2, 2),

so Vi < Vj.

SOME ELEMENTARY GAMES

173

Proof : If either { or i| were something else than we found above, then

m or t respectively must have a component 0, owing to the characteristic

► ►

condition (17 :D) of 17.9. But then rj or £ would differ from the above
values since in these both components are positive. So we see: If either

£ or rj differs from the above values, then both do. And then both must
have a component 0. For both the other component is then 1, i.e. both are

coordinate vectors. 1 Hence the saddle point of K( £ , rj ) which they repre-
sent is really one of 3C(ti, r 2 ), — cf. (17 :E) in 17.9. Thus the game would be
strictly determined; but we know that it is not in this case.

This completes the proof.

All four expressions in (18:11) are now seen to have the same value,
namely

3C(1, 1)3C(2, 2) - 3C(1, 2)3C(2, 1)

JC(1, 1) + 3C(2, 2) - 3C(1, 2) - X(2, 1)

and by (17:5 :a), (17:5 :b) in 17.5.2. this is the value of v'. So we have

(18:13)

3C(1, 1)5C(2, 2) - 3C(1, 2)3C(2, 1)
3C(1, 1) + 0C(2, 2) - 3C(1, 2) - 3C(2, 1)‘

18.3. Qualitative Characterizations

18 . 3 . 1 . The formal results in 18.2. can be summarized in various ways
which make their meaning clearer. We begin with this criterion:

The fields (1, 1), (2, 2) form one diagonal of the matrix scheme of Fig. 27,
the fields (1, 2), (2, 1) form the other diagonal.

We say that two sets of numbers E and F are separated either if every
element of E is greater than every element of F ) or if every element of E
is smaller than every element of F.

Consider now the Cases (A), (B i), (B 2 ) of 18.2. In the first two cases
the game is strictly determined and the elements on one diagonal of the
matrix are not separated from those on the other. 2 In the last case the
game is not strictly determined, and the elements on one diagonal of the
matrix are separated from those on the other. 8

Thus separation of the diagonals is necessary and sufficient for the game
not being strictly determined. This criterion was obtained subject to
the use made in 18.2. of the adjustments of 18.1.3. But the three processes
of adjustment described in 18.1.3. affect neither strict determinateiiess
nor separation of the diagonals. 4 Hence our first criterion is always valid.
We restate it:

Ml, 0) or {0,1).

2 Case (A):3C(1, 1) £ JC(1, 2) £ 3C(2, 2) by (18:2). Case (Bi): JC(1, 1) > JC(1, 2)
£ 3C(2, 2) by (18:7).

8 Case (£*): 3C(1, 1) £ 3C(2, 2) > 3C(1, 2) £ 3C(2, 1) by (18:10).

4 The first is evident since these are only changes in notation, inessential for the game.
The second is immediately verified.

174

ZERO-SUM TWO-PERSON GAMES: EXAMPLES

(18 :A) The game is not strictly determined if and only if the elements

on one diagonal of the matrix are separated from those on the
other.

18 . 3 . 2 . In case (£ 2 ), i.e. when the game is not strictly determined, both

the (unique) £ of A and the (unique) 77 of £ which we found, have both com-
ponents 7 * 0. This, as well as the statement of uniqueness, is unaffected
by adjustments described in 18.1.3. 1 So we have:

(18 :B) If the game is not strictly determined, then there exists only

one good strategy £ (i.e. in A) and only one good strategy 77
(i.e. in 5), and both have both their components positive.

I.e. both players must really resort to mixed strategies.

— > — > — > _ — > _

According to (18 :B) no component of £ or 77 (£ in A, 77 in £) is zero.
Hence the criterion of 17.9. shows that the argument which preceded
(18:11) — which was then sufficient without being necessary — is now
necessary (and sufficient). Hence (18:11) must be satisfied, and therefore
all of its consequences are true. This applies in particular to the values
£ 1 , £ 2 , 771 , 772 given after (18:11), and to the value of v' given in (18:13).
All these formulae thus apply whenever the game is not strictly determined.

18 . 3 . 3 . We now formulate another criterion:

In a general matrix 3C(ri, r 2 ) — cf. Fig. 15 on p. 99 — (we allow for a
moment any /Si, 0 2 ) we say that a row (say t[) or a column (say r 2 ) majorizes
another row (say r") or column (say r' 2 '), respectively, if this is true for their
corresponding elements without exception. I.e. if 3C(rJ, r 2 ) ^ 3C(r'/, r 2 )
for all t 2 , or if 3C(ri, t' 2 ) ^ 3C(ri, t 2 ') for all

This concept has a simple meaning: It means that the choice of t[
is at least as good for player 1 as that of r'/ — or that the choice of r 2 is at
most as good for player 2 as that of r' 2 ' — and that this is so in both cases
irrespective of what the opponent does . 2

Let us now return to our present problem ( 0 i = 0 2 = 2). Consider
again the Cases (A), (£ 1 ), (£ 2 ) of 18.2. In the first two cases a row or a
column majorizes the other . 8 In the last case neither is true . 4

Thus the fact that a row or a column majorizes the other is necessary and
sufficient for r being strictly determined. Like our first criterion this is
subject to the use made in 18.2. of the adjustments made in 18.1.3. And, as
there, those processes of adjustment affect neither strict determinateness
nor majorization of rows or columns. Hence our present criterion too is
always valid. We restate it:

(18 :C) The game T is strictly determined if and only if a row or a

column majorizes the other.

1 These too are immediately verified.

* This is, of course, an exceptional occurrence : In general the relative merits of two
alternative choices will depend on what the opponent does.

’Case (A): Column 1 majorizes column 2, by (18:2) Case (B 1 ): Row 1 majorizes
row 2 by (18:7).

4 Case (B t ): (18:10) excludes all four possibilities, as is easily verified.

SOME ELEMENTARY GAMES

175

18 . 3 . 4 . That the condition of ( 18 :C) is sufficient for strict determinateness
is not surprising: It means that for one of the two players one of his possible
choices is under all conditions at least as good as the other (cf. above). Thus
he knows what to do and his opponent knows what to expect, which is likely
to imply strict determinateness.

Of course these considerations imply a speculation on the rationality
of the behavior of the other player, from which our original discussion is
free. The remarks at the beginning and at the end of 15.8. apply to a
certain extent to this, much simpler, situation.

What really matters in this result (18 :C) however is that the necessity
of the condition is also established; i.e. that nothing more subtle than
outright majorization of rows or columns can cause strict determinateness.

It should be remembered that we are considering the simplest possible
case: 0i = 0 2 = 2. We shall see in 18.5. how conditions get more involved
in all respects when ($ i, £ 2 increase.

18.4. Discussion of Some Specific Games. (Generalized Forms of Matching Pennies)

18 . 4 . 1 . The following are some applications of the results of 18.2. and
18.3.

(a) Matching Pennies in its ordinary form, where the X matrix of Figure
27 is given by Figure 12 on p. 94. We know that this game has the value

v' = 0

and the (unique) good strategies

7=7= in)

(Cf. 17.1. The formulae of 18.2. will, of course, give this immediately.)

18 . 4 . 2 . (b) Matching Pennies, where matching on heads gives a double
premium. Thus the matrix of Figure 27 differs from that of Figure 12 by
the doubling of its (1, 1) element:

1

2

1

2

-1

2

-i

1

Figure 28a.

The diagonals are separated (1 and 2 are > than — 1), hence the good strate-
gies are unique and mixed (cf. (18:A), (18 :B)). By using the pertinent
formulae of case (B 2 ) in 18.2.5., we obtain the value

v' = i

and the good strategies

7 = lit), 7 = it,*}.

It will be observed that the premium put on matching heads has increased
the value of a play for player 1 who tries to match. It also causes him to

176

ZERO-SUM TWO-PERSON GAMES: EXAMPLES

choose heads less frequently, since the premium makes this choice plausible
and therefore dangerous. The direct threat of extra loss by being matched
on heads influences player 2 in the same way. This verbal argument has
some plausibility but is certainly not stringent. Our formulae which yielded
this result, however, were stringent.

18 . 4 . 3 . (c) Matching Pennies, where matching on heads gives a double
premium but failing to match on a choice (by player 1) of heads gives a
triple penalty. Thus the matrix of Figure 27 is modified as follows:

1

2

1

2

-3

2

-1

1

Figure 286.

The diagonals are separated (1 and 2, are > than — 1, —3), hence the good
strategies are unique and mixed (cf. as before). The formulae used before
give the value

v' = - I,

and the good strategies

£ = ih $}, V = {4-> f 1 •

We leave it to the reader to formulate a verbal interpretation of this
result, in the same sense as before. The construction of other examples
of this type is easy along the lines indicated.

18 . 4 . 4 . (d) We saw in 18.1.2. that these variants of Matching Pennies
are, in a way, the simplest forms of zero-sum two-person games. By this
circumstance they acquire a certain general significance, which is further
corroborated by the results of 18.2. and 18.3.: indeed we found there that
this class of games exhibits in their simplest forms the conditions under
which strictly and not-strictly determined cases alternate. As a further
addendum in the same spirit we point out that the relatedness of these
games to Matching Pennies stresses only one particular aspect. Other
games which appear in an entirely different material garb may, in reality,
well belong to this class. We shall give an example of this:

The game to be considered is an episode from the Adventures of Sherlock
Holmes. 1 * 2

1 Conan Doyle: The Adventures of Sherlock Holmes, New York, 1938, pp. 550-551.

* The situation in question is of course again to be appraised as a paradigm of many
possible conflicts in practical life. It was expounded as such by 0. Morgenstern : Wirt-
schaftsprognose, Vienna, 1928, p. 98.

The author does not maintain, however, some pessimistic views expressed id. or in
“ Vollkommene Voraussicht und wirtschaftliches Gleichgewicht,” Zeitschrift fur Nation-
alokonomie, Vol. 6, 1934.

Accordingly our solution also answers doubts in the same vein expressed by K.
Menger: Neuere Fortschritte in den exacten Wissenschaften, “Einige neuere Fort-
schritte in der exacten Behandlung Socialwissenschaftlicher Probleme,” Vienna, 1936,
pp. 117 and 131.

SOME ELEMENTARY GAMES

177

Sherlock Holmes desires to proceed from London to Dover and hence
to the Continent in order to escape from Professor Moriarty who pursues
him. Having boarded the train he observes, as the train pulls out, the
appearance of Professor Moriarty on the platform. Sherlock Holmes
takes it for granted — and in this he is assumed to be fully justified — that
his adversary, who has seen him, might secure a special train and overtake
him. Sherlock Holmes is faced with the alternative of going to Dover or
of leaving the train at Canterbury, the only intermediate station. His
adversary — whose intelligence is assumed to be fully adequate to visualize
these possibilities — has the same choice. Both opponents must choose the
place of their detrainment in ignorance of the others corresponding decision.
If, as a result of these measures, they should find themselves, in fine , on
the same platform, Sherlock Holmes may with certainty expect to be killed
by Moriarty. If Sherlock Holmes reaches Dover unharmed he can make
good his escape.

What are the good strategies, particularly for Sherlock Holifies? This
game has obviously a certain similarity to Matching Pennies, Professor
Moriarty being the one who desires to match. Let him therefore be
player 1, and Sherlock Holmes be player 2. Denote the choice to proceed to
Dover by 1 and the choice to quit at the intermediate station by 2. (This
applies to both r i and t 2 .)

Let us now consider the X matrix of Figure 27. The fields (1, 1) and
(2, 2) correspond to Professor Moriarty catching Sherlock Holmes, which it
is reasonable to describe by a very high value of the corresponding matrix
element, — say 100. The field (2, 1) signifies that Sherlock Holmes suc-
cessfully escaped to Dover, while Moriarty stopped at Canterbury. This is
Moriarty’s defeat as far as the present action is concerned, and' should be
described by a big negative value of the matrix element — in the order of
magnitude but smaller than the positive value mentioned above — say, —50.
The field (1, 2) signifies that Sherlock Holmes escapes Moriarty at the
intermediate station, but fails to reach the Continent. This is best viewed
as a tie, and assigned the matrix element 0.

The X matrix is given by Figure 29:

1

2

1

100

0

2

-50

100

Figure 29.

As in (b), (c) above, the diagonals are separated (100 is > than 0, —50);
hence the good strategies are again unique and mixed. The formulae
used before give the value (for Moriarty)

v' = 40

178

ZERO-SUM TWO-PERSON GAMES: EXAMPLES

and the good strategies ( £ for Moriarty, rj for Sherlock Holmes) :

£ = \i> z)y V = {£, £).

Thus Moriarty should go to Dover with a probability of 60 %, while
Sherlock Holmes should stop at the intermediate station with a probability
of 60 %, — the remaining 40 % being left in each case for the other alternative. 1

18.5. Discussion of Some Slightly More Complicated Games

18 . 6 . 1 . The general solution of the zero-sum two-person game which we
obtained in 17.8. brings certain alternatives and concepts particularly
into the foreground: The presence or absence of strict determinateness, the
value v' of a play, and the sets A, 5 of good strategies. For all these we
obtained very simple explicit characterizations and determinations in 18.2.
These became even more striking in the reformulation of those results in
18.3.

This simplicity may even lead to some misunderstandings. Indeed,
the results of 18.2., 18.3. were obtained by explicit computations of the most
elementary sort. The combinatorial criteria of (18 :A), (18:C) in 18.3.
for strict determinateness were — at least in their final form — also consider-
ably more straightforward than anything we have experienced before.
This may give occasion to doubts whether the somewhat involved consider-
ations of 17.8. (and the corresponding considerations of 14.5. in the case
of strict determinateness) were necessary, — particularly since they are
based on the mathematical theorem of 17.6. which necessitates our analysis
of linearity and convexity in 16. If all this could be replaced by discussions
in the style of 18.2., 18.3. then our mode of discussion of 16. and 17. would be
entirely unjustified. 2

This is not so. As pointed out at the end of 18.3., the great simplicity
of the procedures and results of 18.2. and 18.3. is due to the fact that they
apply only to the simplest type of zero-sum two-person games: the Matching
Pennies class of games, characterized by Pi = 0 2 = 2. For the general
case the more abstract machinery of 16. and 17. seems so far indispensable.

1 The narrative of Conan Doyle — excusably — disregards mixed strategies and states
instead the actual developments. According to these Sherlock Holmes gets out at the
intermediate station and triumphantly watches Moriarty ’s special train going on to
Dover. Conan Doyle* 8 solution is the best possible under his limitations (to pure
strategies), insofar as he attributes to each opponent the course which we found to be the
more probable one (i.e. he replaces 60% probability by certainty). It is, however,
somewhat misleading that this procedure leads to Sherlock Holmes's complete victory,
whereas, as we saw above, the odds (i.e. the value of a play) are definitely iii favor of
— ► — >

Moriarty. (Our result for £ , rj yields that Sherlock Holmes is as good as 48% dead
when his train pulls out from Victoria Station. Compare in this connection the sugges-
tion in Morgenstem , loc. cit., p. 98, that the whole trip is unnecessary because the loser
could be determined before the start.)

1 Of course it would not lack rigor, but it would be an unnecessary use of heavy
mathematical machinery on an elementary problem.

SOME ELEMENTARY GAMES

179

It may help to see these things in their right proportions if we show by
some examples how the assertions of 18.2., 18.3. fail for greater values
of 0 .

18 . 5 . 2 . It will actually suffice to consider games with 0 i = 0 2 = 3.
In fact they will be somewhat related to Matching Pennies, — more general
only by introduction of a third alternative.

Thus both players will have the alternative choices 1 , 2, 3 (i.e. the values
for ti, r 2 ). The reader will best think of the choice 1 in terms of choosing
“heads,” the choice 2 of choosing “tails” and the choice 3 as something like
“calling off.” Player 1 again tries to match. If either player “calls off,”
then it will not matter whether the other player chooses “heads” or “tails,”
— the only thing of importance is whether he chooses one of these two at
all or whether he “calls off” too. Consequently the matrix has now the
appearance of Figure 30:

N \ T 2

Tl \

1

2

3

1

1

-1

7

2

-1

1

7

3

a

a

0

Figure 30.

The four first elements — i.e. the first two elements of the first two rows —
are the familiar pattern of Matching Pennies (cf. Fig. 12). The two fields
with a are operative when player 1 “calls off” and player 2 does not. The
two elements with 7 are operative in the opposite case. The element with
0 refers to the case where both players “ call off.” By assigning appropriate
values (positive, negative or zero) we can put a premium or a penalty
on any one of these occurrences, or make it indifferent.

We shall obtain all the examples we need at this juncture by specializing
this scheme, — i.e. by choosing the above a, 0 , 7 appropriately.

18 . 5 . 3 . Our purpose is to show that none of the results (18:A), (18 :B),
(18 :C) of 18.3. is generally true.

Ad (18 :A): This criterion of strict determinateness is clearly tied to the
special case 0 i = 02 = 2 : For greater values of 0 i, 0 2 the two diagonals do
not even exhaust the matrix rectangle, and therefore the occurrence on the
diagonal alone cannot be characteristic as before.

Ad (18 :B) : We shall give an example of a game which is not strictly
determined, but where nevertheless there exists a good strategy which is
pure for one player (but of course not for the other). This example has the
further peculiarity that one of the players has several good strategies, while
the other has only one.

180

ZERO-SUM TWO-PERSON GAMES: EXAMPLES

We choose in the game of Figure 30 a, 0, 7 as follows:

N \ r *

1

2

3

1

1

-1

0

2

-1

1

0

3

a

a

-a

Figure 31 .

a > 0, 5 > 0. The reader will determine for himself which combinations of
“calling off” are at a premium or are penalized in the previously indicated
sense.

This is a complete discussion of the game, using the criteria of 17.8.

For £ = {i, 0 ) always K( £ , 77 ) = 0 , i.e. with this strategy player 1

cannot lose. Hence v' ^ 0 . For 77 = 8 * = { 0 , 0 , 1 } always K( £ , 77 )
S 0; 1 i.e. with this strategy player 2 cannot lose. Hence v' ^ 0. Thus we
have

v' = 0

Consequently £ is a good strategy if and only if always K( £ , 77 ) ^ 0 and 77

is a good strategy if and only if always K( £ , 77 ) ^ 0. 2 The former is easily
seen to be true if and only if

£1 — £2 —

and the latter if and only if

_ <

Vl Vi - 2 (a + 5 )’

?3 — 0 ,

77 3 — 1 2 t 7 i.

Thus the set A of all good strategies £ contains precisely one element,

and this is not a pure strategy. The set B of all good strategies 77 , on the
other hand, contains infinitely many strategies, and one of them is pure:

namely 77 = 8 1 = {0, 0, 1}.

The sets A, S can be visualized by making use of the graphical repre-
sentation of Figure 21 (cf . Figures 32, 33) :

Ad (18 :C): We shall give an example of a game which is strictly deter-
mined but in which no two rows and equally no two columns majorize each
other. We shall actually do somewhat more.

18 . 5 . 4 . Allow for a moment any fh, p 2 - The significance of the majoriza-
tion of rows or of columns by each other was considered at the end of 18.3.
It was seen to mean that one of the players had a simple direct motive

1 It is actually equal to — $ £*.

* We leave to the reader the simple verbal interpretation of these statements.

SOME ELEMENTARY GAMES

181

for neglecting one of his possible choices in favor of another, — and this
narrowed the possibilities in a way which could be ultimately connected
with strict determinateness.

Specifically: If the row t" is majorized by the row t\ — i.e. if
&( T i> f 2 ) ^ 3C(tj, r 2 ) for all t 2 — then player 1 need never consider the
choice r'/, since t\ is at least as good for him in every contingency. And:
If the column r' 2 ' majorizes the column t' 2 — i.e. if 3C(ti, r 2 ) ^ 3C(n, r 2 ) for all
t 1 — then player 2 need never consider the choice r 2 , since r 2 is at least as good
for him in every contingency. (Cf. loc. cit., particularly footnote 2 on
p. 174. These are of course only heuristic considerations, cf. footnote 1,

p. 182.)

Figure 32.

Now we may use an even more general set-up: If the row r", — i.e.
the player l’s pure strategy corresponding to t" — is majorized by an average

of all rows r\ t" — i.e. by a mixed strategy £ with the component
£ t " = 0 — then it is still plausible to assume that player 1 need never con-
sider the choice of t", since the other r[ are at least as good for him in every
contingency. The mathematical expression of this situation is this:

(18:14:a)

/ 01

X(r'/, t») £ 3C(n, r s )| Ti for all r*

- T!-l

« in S $i , = 0.

The corresponding situation for player 2 arises if the column r, — i.e.

player 2’s pure strategy corresponding to t '( — majorizes an average of all

— >

columns t* ^ t* , — i.e. a mixed strategy rj with the component r?/' = 0 .
The mathematical expression of this situation is this:

(18 :14 :b)

fit

3C(t i, r' 2 ') ^ 2} sc ( T u T t)Vr, for all ti

r,-l

rt in i S fit , n< = 0.

The conclusions are the analogues of the above.

182

ZERO-SUM TWO-PERSON GAMES: EXAMPLES

Thus a game in which (18:14:a) or (18:14:b) occurs, permits of an
immediate and plausible narrowing of the possible choices for one of the
players. 1 * * * * * *

18 . 5 . 5 . We are now going to show that the applicability of (18:14:a),
(18:14:b) is very limited: We shall specify a strictly determined game in
which neither (18:14:a) nor (18:14:b) is ever valid.

Let us therefore return to the class of games of Figure 30. (Pi = = 3).

We choose 0 < a < 1, 0 = 0, 7 = — a:

\ Tj

Tl \

1

2

3

1

1

-1

— a

2

-1

1

— a

3

a

a

0

Figure 34.

The reader will determine for himself which combinations of “ calling off ”
are at a premium or are penalized in the previously indicated sense.

This is a discussion of the game :

The element (3, 3) is clearly a saddle point, so the game is strictly
determined and

v = v' = 0.

It is not difficult to see now (with the aid of the method used in 18.5.3.),

that the set A of all good strategies { as well as the set B of all good strate-

— > — >

gies rj , contains precisely one element: the pure strategy 6 8 = {0, 0, 1).

On the other hand, the reader will experience little trouble in verifying
that neither (18:14:a) nor (18:14 :b) is ever valid here, i.e. that in Figure 34
no row is majorized by any average of the two other rows, and that no
column majorizes any average of the other two columns.

18.6. Chance and Imperfect Information

18 . 6 . 1 . The examples discussed in the preceding paragraphs make it
clear that the role of chance — more precisely, of probability — in a game is
not necessarily the obvious one, that which is directly provided for in the
rules of the game. The games described in Figures 27 and 30 have rules

1 This is of course a thoroughly heuristic argument. We do not need it, since we have

the complete discussions of 14.5. and of 17.8. But one might suspect that it can be used

to replace or at least to simplify those discussions. The example which we are going to

give in the text seems to dispel any such hope.

There is another course which might produce results: If (18:14:a) or (18:14:b) holds,

then a combination of it with 17.8. can be used to gain information about the sets of good

strategies, A and B. We do not propose to take up this subject here.

SOME ELEMENTARY GAMES

183

which do not provide for chance; the moves are personal without excep-
tion. 1 Nevertheless we found that most of them are not strictly determined,
i.e. that their good strategies are mixed strategies involving the explicit
use of probabilities.

On the other hand, our analysis of those games in which perfect informa-
tion prevails showed that these are always strictly determined, — i.e. that
they have good strategies which are pure strategies, involving no probabili-
ties at all. (Cf. 15.)

Thus from the point of view of the played behavior — i.e. of the strate-
gies to be used — the important circumstance is whether the game is strictly
determined or not, and not at all whether it contains any chance moves.

The results of 15. on games in which perfect information prevails
indicate that there exists a close connection between strict determinateness
and the rules which govern the players’ state of information. To establish
this point quite clearly, and in particular to show that the presence of
chance moves is quite irrelevant, we shall now show this: In every (zero-
sum two-person) game any chance move can be replaced by a combination
of personal moves, so that the strategical possibilities of the game remain
exactly the same. It will be necessary to allow for rules involving imperfect
information of the players, but this is just what we want to demonstrate:
That imperfect information comprises (among other things) all possible
consequences of explicit chance moves. 2

18.6.2. Let us consider, accordingly, a (zero-sum two-person) game T,
and in it a chance move 3fTl*. 3 Enumerate the alternatives as usual by
(t k = 1, • • • , a K and assume that their probabilities p< X) , • • • , are

all equal to 1/a*. 4 Now replace 9f!l* by two personal moves am', am''.

1 The reduction of all games to the normalized form shows even more: It proves that
every game is equivalent to one without chance moves, since the normalized form con-
tains only personal moves.

* A direct way of removing chance moves exists of course after the introduction of the
(pure) strategies and the umpire’s choice, as described in 11.1. Indeed — as the last
step in bringing a game into its normalized form — we eliminated the remaining chance
moves by the explicit introduction of expectation values in 11.2.3.

But we now propose to eliminate the chance moves without upsetting the structure
of the game so radically. We shall replace each chance move individually by personal
moves (by two moves, as will be seen), so that their respective roles in determining the
players’ strategies will always remain differentiated and individually appraisable. This
detailed treatment is likely to give a clearer idea of the structural questions involved than
the summary procedure mentioned above.

3 For our present purposes it is irrelevant whether the characteristics of 9TI* depend
upon the previous course of the play or not.

4 This is no real loss of generality. To see this, assume that the probabilities in
question have arbitrary rational values, — say r x /t , • • • , r a Jt (r i, • • • , r a>l and t
integers). (Herein lies an actual restriction — but an arbitrary small one — since any
probabilities can be approximated by rational ones to any desired degree.)

Now modify the chance move 311* so that it has n + • • • -f ra K = t alternatives
(instead of a* ), designated by a K *= 1, • • • , t (instead of <r* • 1, • • • , a*); and so that
each of the first n values of a K has the same effect on the play as <j k = 1, each of the
next rt values of c K the same as <r* =* 2, etc., etc. Then giving all <r' K — 1, • • • , t the
same probability 1/f, has the same effect as giving <r K — 1, • • • , a K the original prob-
abilities ri/t, • • • , r a Jt .

184

ZERO-SUM TWO-PERSON GAMES: EXAMPLES

9fR' and 9fn'' are personal moves of players 1 and 2 respectively. Both have
a K alternatives; we denote the corresponding choices by <r' = 1, * • • , a*
and It is immaterial in which order these moves are

made, but we prescribe that both moves must be made without any infor-
mation concerning the outcome of any moves (including the other move
9Tl', 9Tl'')* We define a function S(<t', <t") by this matrix scheme. (Cf.
Figure 35. The matrix element is $(</, a"). 1 ). The influence of 9fTl(, 9Tl'' —
i.e. of the corresponding (personal) choices a', <r'' — on the outcome of the
game is the same as that of 3TC* would have been with the corresponding
(chance) choice <r« = 6(<r', a"). We denote this new game by V*. We
claim that the strategical possibilities of T* are the same as those of T.

Figure 35.

18.6.3. Indeed: Let player 1 use in T* a given mixed strategy of T with
the further specification concerning the move 9Tl', 2 to choose all a' = 1,
* • • , a* with the same probability l/a K . Then the game T* — with this
strategy of player 1 — will be from player 2’s point of view the same as T.
This is so because any choice of his at 9Tl'' (i.e. any <r" = 1, • • • , a«)
produces the same result as the original chance move 9Tl«: One look at
Figure 35 will show that the a" = <r'' column of that matrix contains every
number <r = 6(<r', <r") = 1, • • • , a K precisely once, — i.e. that 5(<r', <r")
will assume every value 1, • • • , a K (owing to player Us strategy) with the
same probability 1/a*, just as 9TC* would have done. So from player Us
point of view, T* is at least as good as T.

The same argument with players 1 and 2 interchanged — hence with
the rows of the matrix in Figure 35 playing the roles which the column
had above — shows that from player 2 ; s point of view too T* is at least as
good as T.

1 Arithmetically

*(*>"){ Zs

<r" + 1
*" + 1 + a*

for

for

a' £ <r"
a' <<r"

Hence 5(<r', <r") is always one of the numbers 1, • • • , a*.

* SHli is his personal move, so his strategy must provide for it in r*. There was no
need for this in r, since was a chance move.

SOME ELEMENTARY GAMES

185

Since the viewpoints of the two players are opposite, this means that
T* and T are equivalent . 1

18.7. Interpretation of This Result

18 . 7 . 1 . Repeated application to all chance moves of T, of the operation
described in 18.6.2., 18.6.3., will remove them all, — thus proving the final
contention of 18.6.1. The meaning of this result may be even better under-
stood if we illustrate it by some practical instances of this manipulation.

(A) Consider the following quite elementary “game of chance.” The
two players decide, by a 50%-50% chance device, who pays the other one
unit. The application of the device of 18.6.2., 18.6.3. transforms this game,
which consists of precisely one chance move, into one of two personal moves.
A look at the matrix of Figure 35 for a K = 2 — with the $(</, <j") values 1, 2
replaced by the actual payments 1, —1 — shows that it coincides with
Figure 12. Remembering 14.7.2., 14.7.3. we see that this means — what is
plain enough directly — that this is the game of Matching Pennies.

I.e. : Matching Pennies is the natural device to produce the probabilities
i, i by personal moves and imperfect information. (Recall 17.1.!)

(B) Modify (A) so as to allow for a “tie”: The two players decide by a
33^%, 33£%, 33^% chance device who pays the other one unit, or whether
nobody pays anything at all. Apply again the device of 18.6.2., 18.6.3.
Now the matrix of Figure 35 with a K = 3 — with the 5(o*', <r”) values 1, 2, 3
replaced by the actual payments 0, 1, — 1 — coincides with Figure 13. By
14.7.2., 14.7.3. we see that this is the game of Stone, Paper, Scissors.

I.e., Stone, Paper, Scissors is the natural device to produce the proba-
bilities i, i, i by personal moves and incomplete information. (Recall
17.1.!)

18 . 7 . 2 . (C) The 6(<r', <t”) of Figure 35 can be replaced by another func-
tion, and even the domains a[ — 1, • • • , a« and <r” = 1, • * • , a K by other
domains <r' = 1 , * • • , a' and <r” = 1 , • • • , a”, provided that the follow-
ing remains true: Every column of the matrix of Figure 35 contains each
number 1 ,••*,<*« the same number of times, 2 * * * * and every row contains
each number 1 , • • • , a, the same number of times. 8 Indeed, the con-
siderations of 18.6.2. made use of these two properties of d(a[, <r”) (and of

«") only.

It is not difficult to see that the precaution of “cutting” the deck before
dealing cards falls into this category. When one of the 52 cards has to be
chosen by a chance move, with probability fa, this is usually achieved by
“mixing” the deck. This is meant to be a chance move, but if the player
who mixes is dishonest, it may turn out to be a “personal” move of his.

1 We leave it to the reader to cast these considerations into the precise formalism of 1 1.

and 17.2., 17.8.: This presents absolutely no difficulties, but it is somewhat lengthy.

The above verbal arguments convey the essential reason of the phenomenon under con-

sideration in a clearer and simpler way — we hope.

* Hence aja t times; consequently a K must be a multiple of a*.

1 Hence a ”/«* times; consequently a" must be a multiple of a*.

186 ZERO-SUM TWO-PERSON GAMES: EXAMPLES

As a protection against this, the other player is permitted to point out the
place in the mixed deck, from which the card in question is to be taken, by
“cutting” the deck at that point. This combination of two moves — even
if they are personal — is equivalent to the originally intended chance move.
The lack of information is, of course, the necessary condition for the effec-
tiveness of this device.

Here a, = 52, a' = 52! = the number of possible arrangements of the
deck, a" = 52 the number of ways of “cutting.” We leave it to the reader
to fill in the details and to choose the $(<r', a") for this set-up. 1

19. Poker and Bluffing
19.1. Description of Poker

19.1.1. It has been stressed repeatedly that the case fii = P* = 2 as
discussed in 18.3. and more specifically in 18.4., comprises only the very
simplest zero-sum two-person games. We then gave in 18.5. some instances
of the complications which can arise in the general zero-sum two-person
game, but the understanding of the implications of our general result
(i.e. of 17.8.) will probably gain more by the detailed discussion of a special
game of the more complicated type. This is even more desirable because
for the games with = 0 2 = 2 the choices of then, r 2 , called (pure) strate-
gies, scarcely deserve this name: just calling them “moves” would have
been less exaggerated. Indeed, in these extremely simple games there
could be hardly any difference between the extensive and the normalized form ;
and so the identity of moves and strategies, a characteristic of the normalized
form, is inescapable in these games. We shall now consider a game in the
extensive form in which the player has several moves, so that the passage
to the normalized form and to strategies is no longer a vacuous operation.

19.1.2. The game of which we give an exact discussion is Poker. 2 How-
ever, actual Poker is really a much too complicated subject for an exhaustive
discussion and so we shall have to subject it to some simplifying modifica-

1 We assumed that the mixing is used to produce only one card. If whole “ hands"
are dealt, “cutting” is not an absolute safeguard. A dishonest mixer can produce corre-
lations within the deck which one “cut” cannot destroy, and the knowledge of which
gives this mixer an illegitimate advantage.

1 The general considerations concerning Poker and the mathematical discussions of
the variants referred to in the paragraphs which follow, were carried out by J. von
Neumann in 1926-28, but not published before. (Cf . a general reference in “ Zur Theorie
der Gesellschaftsspiele,” Math. Ann., Vol. 100 [1928]). This applies in particular to the
symmetric variant of 19.4.-19.10., the variants (A), (B) of 19.11.-19.13., and to the entire
interpretation of “Bluffing” which dominates all these discussions. The unsymmetric
variant (C) of 19.14.-19.16. was considered in 1942 for the purposes of this publication.

The work of E. Borel and J, Ville , referred to in footnote 1 on p. 154, also con-
tains considerations on Poker (Vol. IV, 2: “Applications aux Jeux de Hasard,” Chap.
V: “Le jeu de Poker”). They are very instructive, but mainly evaluations of prob-
abilities applied to Poker in a more or less heuristic way, without a systematic use of any
underlying general theory of games.

A definite strategical phase of Poker (“La Relance” — “The Overbid”) is analyzed
on pp. 91-97 loc. cit. This may be regarded also as a simplified variant of Poker, —

POKER AND BLUFFING

187

tions, some of which are, indeed, quite radical. 1 It seems to us, neverthe-
less, that the basic idea of Poker and its decisive properties will be conserved
in our simplified form. Therefore it will be possible to base general con-
clusions and interpretations on the results which we are going to obtain
by the application of the theory previously established.

To begin with, Poker is actually played by any number of persons, 2
but since we are now in the discussion of zero-sum two-person games, we
shall set the number of players at two.

The game of Poker begins by dealing to each player 5 cards out of a
deck. 8 The possible combinations of 5 which he may get in this way —
there are 2,598,960 of them 4 — are called “hands” and arranged in a linear
order, i.e. there is an exhaustive rule defining which hand is the strongest
of all, which is the second, third, • • • strongest down to the weakest. 5
Poker is played in many variants which fall into two classes: “Stud” and
“Draw” games. In a Stud game the player’s hand is dealt to him in its
entirety at the very beginning, and he has to keep it unchanged throughout
the entire play. In “Draw” games there are various ways for a player to
exchange all or part of his hand, and in some variants he may get his hand
in several successive stages in the course of the play. Since we wish to
discuss the simplest possible form, we shall examine the Stud game only.

In this case there is no point in discussing the hands as hands, i.e. as
combinations of cards. Denoting the total number of hands by S —
S = 2,598,960 for a full deck, as we saw — we might as well say that each

comparable to the two which we consider in 19.4.-19.10. and 19.14-19.16. It is actually
closely related to the latter.

The reader who wishes to compare these two variants, may find the following indica-
tions helpful:

(I) Our bids a, b correspond to 1 + a, 1 loc. cit.

(II) The difference between our variant of 19.4.-19.10. and that in loc. cit. is this:
If player 1 begins with a “low" bid, then our variant provides for a comparison of hands,
while that in loc. cit. makes him lose the amount of the “low" bid unconditionally. I.e.
we treated this initial “low" bid as “seeing" — cf. the discussion at the beginning of
19.14., particularly footnote 1 on p. 211 — while in loc. cit. it is treated as “passing."
We believe that our treatment is a better approximation to the phase in question in real
Poker; and in particular that it is needed for a proper analysis and interpretation of
“Bluffing." For technical details cf. footnote 1 on p. 219.

1 Cf. however 19.11. and the end of 19.16.

2 The “optimum" — in a sense which we do not undertake to interpret — is supposed
to be 4 or 5.

3 This is occasionally a full deck of 52 cards, but for smaller numbers of participants
only parts of it — usually 32 or 28 — are used. Sometimes one or two extra cards with
special functions, “jokers," are added.

4 This holds for a full deck. The reader who is conversant with the elements of com-
binatorics will note that this is the number of “combinations without repetitions of
5 out of 52”:

/52\ ^ 52 • 51 • 50 • 49 • 48
\ 5 / * 1 • 2 • 3 • 4 • 5

2,598,960.

* This description involves the well known technical terms “Royal Flush," “Straight
Flush," “Four of a Kind," “Full House," etc. There is no need for us to discuss them
here.

188

ZERO-SUM TWO-PERSON GAMES: EXAMPLES

player draws a number s = 1, • • • , S instead. The idea is that s = S
corresponds to the strongest possible hand, s = S — 1 to the second
strongest, etc., and finally s = 1 to the weakest. Since a “ square deal”
amounts to assuming that all possible hands are dealt with the same prob-
ability, we must interpret the drawing of the above number s as a chance
move, each one of the possible values s = 1, • • • , S having the same
probability 1/S. Thus the game begins with two chance moves: The
drawing of the number $ for player 1 and for player 2, 1 which we denote
by Si and Sj.

19 . 1 . 3 . The next phase of the general game of Poker consists of the
making of “Bids” by the players. The idea is that after one of the players
has made a bid, which involves a smaller or greater amount of money, his
opponent has the choice of “Passing,” “Seeing,” or “Overbidding.”
Passing means that he is willing to pay, without further argument, the
amount of his last preceding bid (which is necessarily lower than the present
bid). In this case it is irrelevant what hands the two players hold. The
hands are not disclosed at all. “Seeing” means that the bid is accepted:
the hands will be compared and the player with the stronger hand receives
the amount of the present bid. “Seeing” terminates the play. “Overbid-
ding” means that the opponent counters the present bid by a higher one,
in which the roles of the players are reversed and the previous bidder has
the choice of Passing, Seeing or Overbidding, etc. 2

19.2. Bluffing

19 . 2 . 1 . The point in all this is that a player with a strong hand is likely
to make high bids — and numerous overbids — since he has good reason to
expect that he will win. Consequently a player who has made a high bid,
or overbid, may be assumed by his opponent — a posteriori! — to have a
strong hand. This may provide the opponent with a motive for “ Passing.”
However, since in the case of “Passing” the hands are not compared, even
a player with a weak hand may occasionally obtain a gain against a stronger
opponent by creating the (false) impression of strength by a high bid, or
by overbid, — thus conceivably inducing his opponent to pass.

This maneuver is known as “Bluffing.” It is unquestionably prac-
ticed by all experienced players. Whether the above is its real motivation
may be doubted; actually a second interpretation is conceivable. That is
if a player is known to bid high only when his hand is strong, his opponent is
likely to pass in such cases. The player will, therefore, not be able to collect
on high bids, or on numerous overbids, in just those cases where his actual
strength gives him the opportunity. Hence it is desirable for him to create

1 In actual Poker the second player draws from a deck from which the first player’s
hand has already been removed. We disregard this as we disregard some other minor
complications of Poker.

2 This scheme is usually complicated by the necessity of making unconditional pay-
ments, the “ante,” at the start, — in some variants for the first bidder, in others for all
those who wish to participate, again in others extra payments are required for the privi-
lege of drawing, etc. We disregard all this.

POKER AND BLUFFING 189

uncertainty in his opponent’s mind as to this correlation, — i.e. to make it
known that he does occasionally bid high on a weak hand.

To sum up: Of the two possible motives for Bluffing, the first is the
desire to give a (false) impression of strength in (real) weakness; the second
is the desire to give a (false) impression of weakness in (real) strength.
Both are instances of inverted signaling (cf. 6.4.3.), — i.e. of misleading the
opponent. It should be observed however that the first* type of Bluffing
is most successful when it “ succeeds,” i.e. when the opponent actually
“ passes,” since this secures the desired gain; while the second is most
successful when it “ fails,” i.e. when the opponent “sees,” since this will
convey to him the desired confusing information. 1

19 . 2 . 2 . The possibility of such indirectly motivated — hence apparently
irrational — bids has also another consequence. Such bids are necessarily
risky, and therefore it can conceivably be worth while to make them riskier
by appropriate counter measures, — thus restricting their use by the oppon-
ent. But such counter measures are ipso facto also indirectly motivated
moves.

We have expounded these heuristic considerations at such length
because our exact theory makes a disentanglement of all these mixed
motives possible. It will be seen in 19.10. and in 19.15.3., 19.16.2. how the
phenomena which surround Bluffing can be understood quantitatively,
and how the motives are connected with the main strategic features of the
game, like possession of the initiative, etc.

19.3. Description of Poker (Continued)

19 . 3 . 1 . Let us now return to the technical rules of Poker. In order to
avoid endless overbidding the number of bids is usually limited. 2 In order
to avoid unrealistically high bids — with hardly foreseeable irrational effects
upon the opponent — there are also maxima for each bid and overbid.
It is also customary to prohibit too small overbids; we shall subsequently
indicate what appears to be a good reason for this (cf. the end of 19.13.).
We shall express these restrictions on the size of bids and overbids in the
simplest possible form: We shall assume that two numbers a, b

a > b > 0

1 At this point we might be accused once more of disregarding our previously stated
guiding principle; the above discussion obviously assumes a series of plays (so that
statistical observation of the opponent's habits is possible) and it has a definitely
“ dynamical" character. And yet we have repeatedly professed that our considerations
must be applicable to one isolated play and also that they are strictly statical.

We refer the reader to 17.3., where this apparent contradiction has been carefully
examined. Those considerations are fully valid in this case too, and should justify our
procedure. We shall add now only that our inconsistency — the use of many plays and
of a dynamical terminology — is a merely verbal one. In this way we were able to make
our discussions more succinct and more akin to the way that these things are talked about
in everyday language. But in 17.3. it was elaborated how all these questionable pictures
can be replaced by the strictly static problem of finding a good strategy.

1 This is the stop rule of 7.2.3.

190

ZERO-SUM TWO-PERSON GAMES: EXAMPLES

are given ab initio , and that for every bid there are only two possibilities:
the bid may be “high,” in which case it is a; or “low,” in which case it is
b . By varying the ratio a/b — which is clearly the only thing that matters —
we can make the game risky when a/b is much greater than 1, or relatively
safe when a/b is only a little greater than 1 .

The limitation of the number of bids and overbids will now be used for a
simplification of the entire scheme. In the actual play one of the players
begins with the initial bid; after that the players alternate.

The advantage or disadvantage contained in the possession of the
initiative by one player — but concurrent with the necessity of acting first! —
constitutes an interesting problem in itself. We shall discuss an (unsym-
metric) form of Poker where this plays a role in 19.14., 19.15. But we
wish at first to avoid being saddled ydth this problem too. In other words,
we wish to avoid for the moment all deviations from symmetry, so as to
obtain the other essential features of Poker in their purest and simplest
form. We shall therefore assume that the two players both make initial
bids, each one ignorant of the others choice. Only after both have made
this bid is each one informed of what the other did, i.e. whether his bid was
“high” or “low.”

19 . 3 . 2 . We simplify further by giving to the players only the choice of
“Passing” or “Seeing,” i.e. by excluding “Overbidding.” Indeed, “ Over-
bidding” is only a more elaborate and intensive expression of the tendency
which is already contained in a high initial bid. Since we wish to do things
as simply as possible, we shall avoid providing several channels for the same
tendency. (Cf. however (C) in 19.11. and 19.14., 19.15.).

Accordingly we prescribe these conditions: Consider the moment
when both players are informed of each other's bids. If it then develops
that both bid “high” or that both bid “low,” then the hands are compared
and the player with the stronger hand receives the amount a or b respectively
from his opponent. If their hands are equal, no payment is made. If on
the other hand one bids “high” and one bids “low,” then the player with
the low bid has the choice of “Passing” or “Seeing.” “Passing” means
that he pays to the opponent the amount of the low bid (without any
consideration of their hands). “Seeing” means that he changes over
from his “low” bid to the “high” one, and the situation is treated as if
they both had bid “high” in the first place.

19.4. Exact Formulation of the Rules

19 . 4 . We can now sum up the preceding description of our simplified
Poker by giving an exact statement of the rules agreed upon:

First, by a chance move each player obtains his “hand,” a number
s = 1, • • • $, each one of these numbers having the same probability l/S.
We denote the hands of players 1, 2, by s u s 2 respectively.

After this each player will, by a personal move, choose either a or fe,
the “high” or “low” bid. Each player makes his choice (bid) informed

POKER AND BLUFFING

191

about his own hand, but not about his opponent's hand or choice (bid).
Lastly, each player is informed about the other's choice but not about his
hand. (Each still knows his own hand and choice.) If it turns out that
one bids “high" and the other “low," then the latter has the choice of
‘‘Seeing" or “Passing."

This is the play. When it is concluded the payments are made as
follows: If both players bid “high," or if one bids “high," and the other

bids “low" but subsequently “Sees," then for s x = $ 2 player 1 obtains

<

a

from player 2 the amount 0 respectively. If both players bid “low,"

— a

> b

then for 8i = s 2 player 1 obtains from player 2 the amount 0 respectively.
< -b

If one player bids “high," and the other bids “low" and subsequently
“Passes," then the “high bidder" being ]>> player 1 obtains from player 2

the amount

b 1
- 6 *

19.5. Description of the Strategies

19 . 5 . 1 . A (pure) strategy in this game consists clearly of the following
specifications : To state for every “ hand " s = 1, • • • , S whether a “ high "
or a “low" bid will be made, and in the latter case the further statement
whether, if this “low" bid runs into a “high" bid of the opponent, the player
will “See" or “Pass." It is simpler to describe this by a numerical index
i $ = 1, 2, 3; i 9 = 1 meaning a “high" bid; i, = 2 meaning a “low" bid
with subsequent “Seeing" (if the occasion arises); i, = 3 meaning a “low"
bid with subsequent “Passing" (if the occasion arises). Thus the strategy
is a specification of such an index i B for every $ = 1, • • • , S, — i.e. of the
sequence i\ } * • • , is-

This applies to both players 1 and 2. Accordingly we shall denote the
above strategy by 2i(ii, • • * is) or S 2 ( ji, * * ' > js ).

Thus each player has the same number of strategies: as many as there
are sequences ii, • • • , i s , — i.e. precisely 3 s . With the notations of 11.2.2.

Pi = ft = fi = 3 s .

1 For the sake of absolute formal correctness this should still be arranged according
to the patterns of 6. and 7. in Chapter II. Thus the two first-mentioned chance moves
(the dealing of hands) should be called moves 1 and 2; the two subsequent personal iqoves
(the bids), moves 3 and 4; and the final personal move (“Passing” or “Seeing”), move 5.

In the case of move 5, both the player whose personal move it is, and the number
of alternatives, depend on the previous course of the play as described in 7.1.2. and 9.1.5.
(If both players bid “high” or both bid “low,” then the number of alternatives is 1, and
it does not matter to which player we ascribe this vacuous personal move. If one bids
“high” and the other bids “low,” then the personal move is the “low” bidder's).

A consistent use of the notations loc. cit. would also necessitate writing <n, <r* for
*i, *s;<ri, 0 * 4 for the “high” or “low” bid;<r # for “Passing” or “Seeing.”

We leave it to the reader to iron out all these differences.

192

ZERO-SUM TWO-PERSON GAMES: EXAMPLES

If we wanted to adhere rigorously to the notations of loc. cit., we should
now enumerate the sequences i h • • • , is with a n = 1, • • • , 0 and then
denote the (pure) strategies of the players 1, 2 by 2^, 2^». But we prefer
to continue with our present notations.

We must now express the payment which player 1 receives if the strate-
gies 2i(ii, • • • , i s ), 2 2 (ji, * • • , js) are used by the two players. This
is the matrix element 3C(t\, • • • , i s \ji, • • • , js). 1

If the players have actually the “hands” si, s 2 then the payment
received by player 1 can be expressed in this way (using the rules stated
above): It is £ 90n{$ _. #j) (t v j. t ) where sgn(si — s 2 ) is the sign of Si — $ 2 , 2 and
where the three functions

£+(i> j)i &o (i, j ), £-(i> j ) i , j — 1, 2, 3.

can be represented by the following matrix schemes: 3

\j

i \

1

2

3

1

a

a

6

2

a

b

b

3

-6

b

b

£+(*> j)

Figure 36.

Now Si, 82 originate

\j

i \

1

2

3

1

0

0

b

2

0

0

0

3

-b

0

0

•Co (i,j)
Figure 37.

\ j

i \

1

2

3

1

—a

—a

b

2

~ a

—b

-6

3

-b

-b

- b

Figure 38.

from chance moves, as described above. Hence

8

*^(iif * * * f is\ji) • * * , js) = ^ j*,)* 4

19.5.2. We now pass to the (mixed) strategies in the sense of 17.2.

— > — ►

These are the vectors £ , 77 belonging to Sp. Considering the notations
1 The entire sequence ti, • • • , is is the row index, and the^en tire sequence j lt • • • ,
js is the column index. In our original notations the strategies were Sf*, Sj* and the
matrix element 3C(n, r*).

+ >

* I.e. 0 for «i — 82 respectively. It expresses in an arithmetical form which hand is
stronger.

* The reader will do well to compare these matrix schemes with our verbal statements
of the rules, and to verify their appropriateness.

Another circumstance which is worth observing is that the symmetry of the game
corresponds to the identities

•C+(i, j) » £ 0 (i,j) = “£oO*»t)

4 The reader may verify

3C(ii, * • • , is\j\, , js) 3C(ji» • • • , Jfljii, • * * , is)
as a consequence of the relations at the end of footnote 3 above. I.e.

3C(ii, • • • , is\ji, • • • ,js)

is skew-symmetric, expressing once more the, symmetry of the game.

POKER AND BLUFFING 193

which we are now using, we must index the components of these vectors

also in the new way: We must write £ ti j instead of $ r f rj Tt .

We express (17:2) of 17.4.1., which evaluates the expectation value of
player Us gain

K (£> v) = ^ 3C (/,, • • • ia\ji, • • • t ja)ki t

»i is

= Si ^ j’Jb, i.

*1 *sJi Ja *1.*J

There is an advantage in interchanging the two 2 and writing

K( £ > *7 ) = ^2 ^ ^ £ s 0n ($ r $ t )(i 9l1 j* t )£ x x i g rji l j 8 -

»i» . . . ,tsJi Ja

If we now put

(19:1)

£*

»i, . . . ,t a ‘excluding i fj

t - «

(19:2)

% i.

ii, ■ ■ Ja excluding j lf

i..-i

then the above equation becomes

(19:3) K( £ , n ) = gj ^ ^ j)pX>.

It is worth while to expound the meaning of (19:1)-(19:3) verbally.
(19:1) shows that pji is the probability that player 1, using the mixed

strategy £ , will choose i when his “hand” is $ij (19:2) shows that <rj* is the

— ►

probability that player 2, using the mixed strategy tj , will choosey when his
“hand” is S 2. 1 Now it is intuitively clear that the expectation value

K( £ , rj ) depends on these probabilities pji, only, and not on the underly-
ing probabilities rj ii themselves. 2 The formula (19:3) can

1 We know from 19 4. that i or j — 1 means a “high” bid, i — 2, 3 a “low” bid with
(the intention of) a subsequent “Seeing” or “Passing” respectively.

2 This means that two different mixtures of (pure) strategies may in actual effect be
the same thing.

Let us illustrate this by a simple example. Put S — 2, i.e. let there be only a “high ”
and a “low” hand. Consider i » 2, 3 as one thing, i.e. let there be only a “high” and a

194

ZERO-SUM TWO-PERSON GAMES: EXAMPLES

easily be seen to be correct in this direct way: It suffices to remember
the meaning of the £ tgn ^ r9t) (i t j ) and the interpretation of the pji, < 7 y».

19 . 6 . 3 . It is clear, both from the meaning of the pji, o-J i and from their
formal definition (19:1), (19:2), that they fulfill the conditions

(19:4) all pf ^ 0, | pj. = 1

t-i

3

(19:5) all <r** ^0, £ o-J* = 1

;-i

On the other hand, any pj», <rj» which fulfill these conditions can be obtained

from suitable £ , tj by (19:1), (19:2). This is clear mathematically , 1 and
intuitively as well. Any such system of pji, is one of probabilities which
define a possible modus procedendi , — so they must correspond to some mixed
strategy.

(19:4), (19:5) make it opportune to form the 3-dimensional vectors

P * 1 = {pi 1 , p*S P3 1 !, a 8 > = {*!*, <rj*, o-J*}.

— > — >

Then (19:4), (19:5) state precisely that all p* 1 , cr'> belong to /S 3 .

This shows how much of a simplification the introduction of these

— > — >

vectors is: £ (or jj ) was a vector in Sp, i.e. depending on /3 — 1 = 3 s — 1

> >

numerical constants; the p a i (or the (7 * 2 ) are S vectors in £ 3 , i.e. each one
depends on 2 numerical constants; hence they amount together to 2 S
numerical constants. And 3 s - 1 is much greater than 25, even for moder-
ate S . 2

“low” bid. Then there are four possible (pure) strategies, to which we shall give names:

“Bold”: Bid “high” on every hand.

“Cautious”: Bid “low” on every hand.

“Normal”: Bid “high” on a “high” hand, “low” on a “low” hand.

“Bluff”: Bid “high” on a “low” hand, “low” on a “high” hand.

Then a 50-50 mixture of “Bold” and “Cautious” is in effect the same thing as a
50-50 mixture of “Normal” and “Bluff”: both mean that the player will — according
to chance — bid 50-50 “high” or “low” on any hand.

Nevertheless these are, in our present notations, two different “mixed” strategies,

i.e. vectors £ .

This means, of course, that our notations, which were perfectly suited to the general
case, are redundant for many particular games. This is a frequent occurrence in mathe-
matical discussions with general aims.

There was no reason to take account of this redundance as long as we were working
out the general theory. But we shall remove it now for the particular game under
consideration.

1 Put e.g. Sq ig * p} t • . . . • p? a , r jj i )g * * l h ' . . . -<r, s and verify the (17:1 :a),

(17:1 :b) of 17.2.1. as consequences of the above (19:4), (19:5).

* Actually S is about 2i millions (cf. footnote 4 on p. 187); so 3 s — 1 and 2 S are both
great, but the former is quite exorbitantly the greater.

POKER AND BLUFFING

195

19.6. Statement of the Problem

19 . 6 . Since we are dealing with a symmetric game, we can use the char-

acterization of the good (mixed) strategies — i.e. of the £ in A — given in

— ^

(17:H) of 17.11.2. It stated this: £ must be optimal against itself, — i.e.
Min— K( £ , rj) must be assumed for rj = £ .

Now we saw in 19.5. that K( £ , rj ) depends actually on the p *», a *«. So

we may write for it, K( p l , • • • , p s | a l , • • • a 8 ). Then (19:3) in 19.5.2.
states (we rearrange the 2 somewhat)

(19:6) K(7‘, • • • , 7 S \7\ • ■ • , 7 s ) - j) P ‘X'.

•l.< •»}

— ► — >

And the characteristic of the p l , • • • , p 8 of a good strategy is that

Min-

K(p l ,

is assumed at a 1 = p x , • • • , a 8 = p s . The explicit conditions for this
can be found in essentially the same way as in the similar problem of 17.9. 1. ;
we will give a brief alternative discussion.

The Min— — of (19:6) amounts to a minimum with respect to

a 1 , . . . , a s

each c r *, • • * , a 8 separately. Consider therefore such a <r •*. It is
restricted only by the requirement to belong to S 3 , — i.e. by

3

all £ 0, £ <7*« = 1.

(19:6) is a linear expression in these three components a\', < 75 *, crj«. Hence

— ►

it assumes its minimum with respect to <7 ** there where all those components
< 7 ji which do not have the smallest possible coefficient (with respect to j,
cf. below), vanish.

The coefficient of <r** is

<£. c «<. r .,)(*, j)p*‘
Thus (l9 :6) becomes

to be denoted by ^ 7J1.

196

ZERO-SUM TWO-PERSON GAMES: EXAMPLES

And the condition for the minimum (with respect to a •■) is this:

(19:8) For each pair s 2 , j, for which 7 J* does not assume its minimum

(in j l ), we have <rj* = 0 .

Hence the characteristic of a good strategy — minimization at a 1 = p l , • • • ,
— ► — >

<r 8 = p 5 — is this:

(19 :A) p • • • , p 5 describe a good strategy, i.e. a { in A, if and

only if this is true:

For each pair s 2 , j for which 7 J* does not assume its minimum
(in j *), we have pj* = 0 .

We finally state the explicit expressions for the 7 **, of course by using the
matrix schemes of Figures 36-38. They are

(19:9 :a) 7i* = g | ^ ( — opf 1 ~ ap 2 l - &P 3 O — &Pa*

«i»i

s

+ ^ («Pi l + apj 1 ~ &p;oJ>

(19:9:b) 7 * 2 * = g { ^ (-apf 1 ~ bp 2 ‘ ~ hpj 1 )

«.-i

5

+ ^ (apfi H- fep'i + bpj.oj^

«,-i

(19:9:c) 7 ** = g j ^ (&P 1 1 - bp 2 l - bp| l ) + bp**

s

+ ^ (&Pi» + bp\i + 6pJ«) |

• 1 - 1

19.7. Passage from the Discrete to the Continuous Problem

19 . 7 . 1 . The criterion (19:A) of 19.6., together with the formulae (19:7),
(19:9:a), (19:9 :b), (19:9:c), can be used to determine all good strategies . 2
This discussion is of a rather tiresome combinatorial character, involving
the analysis of a number of alternatives. The results which are obtained

1 We mean in j and not in «>, j\

* This determination has been carried out by one of us and will be published elsewhere.

POKER AND BLUFFING

197

are qualitatively similar to those ‘which we shall derive below under some-
what modified assumptions, except for certain differences in very delicate
detail which may be called the “fine structure” of the strategy. We shall
say more about this in 19.12.

For the moment we are chiefly interested in the main features of the solu-
tion and not in the question of “fine structure. ” We begin by turning our
attention to the “granular” structure of the sequence of possible hands
s = 1, • • • , S.

If we try to picture the strength of all possible “hands” on a scale from
0 % to 100%, or rather of fractions from 0 to 1, then the weakest possible
hand, 1, will correspond to 0, and the strongest possible hand, S } to 1.

8 — 1 .

Hence the “hand” $(= 1, • • • , S) should be placed at z = on

scale. I.e. we have this correspondence:

Possible 1
“hands”)

Old scale :

s =

1

KB

II

New scale:

n

1

H

B

1

1

z —

u

Figure 39.

Thus the values of z fill the interval
(19:10) 0 ^ z g 1

very densely, 1 but they form nevertheless a discrete sequence. This is the
“granular” structure referred to above. We will now replace it by a
continuous one.

I.e. we assume that the chance move which chooses the hand s — i.e. z —
may produce any z of the interval (19:10). We assume that the probability
of any part of (19:10) is the length of that part, i.e. that z is equidistributed
over (19:10). 2 We denote the “hands” of the two players 1, 2 by Zi, z 2
respectively.

19.7.2. This change entails that we replace the vectors

p •», a ($i, s 2} = 1, * * • , S) by vectors p f >, a •* (0 ^ z h z 2 ^ 1); but
they are, of course, still probability vectors of the same nature as before,
i.e. belonging to S 2 . In consequence, the components (probabilities)
*5* ($i, s 2 = 1, * • • , S; j = 1, 2, 3) give way to the components
P*S (0 ^ z h z 2 ^ 1 ;i,j = 1, 2, 3). Similarly the yj* (in (19:9:a), (19:9:b),
(19:9:c) of 19.6.) become 7J*.

We now rewrite the expressions for K and the 7* in the formulae (19:7),
(19:9:a), (19:9:b), (19:9:c) in 19.6. Clearly all sums

1 It will be remembered (cf. footnote 4 on p. 187) that, <8 is about 2J millions.

* This is the so-called geometrical probability.

198

ZERO-SUM TWO-PERSON GAMES: EXAMPLES

5 S

J2-J2

*i — 1 1

must be replaced by integrals

/ 1 • • • dz u I 1 • • • dz 2}

Jo Jo

sums

«,-i s

by integrals

[" ■ ■ ■ dzi, f 1 ■ ■ ■ dz u

JO J z,

while isolated terms behind a factor 1/S may be neglected . 1 - 2 These being
understood, the formulae for K and the y) (i.e. 7 *) become:

(19:7*) K = X / 0 ‘ W*.

y

(19:9:a*) 7 {« = f *’ (— apj 1 — apf 1 — bpfOdzi + f 1 (apl> + ap}> — bp}‘)dzi,
Jo Jt,

(19:9:b*) 7 ,* = [‘ ' (— ap{< — — bpl<)dzi + f 1 (ap{> + 6 p|> + bpi‘)dz u

Jo Jt x

(19:9:c*) 7 J* = f * 1 ( 6 pJ» — bp^ — bp\')dzi + f l ( bp \ 1 + + bptfdzi.

Jo Jt t

And the characterization (19 :A) of 19.6. goes over into this:

(19 :B) The p* (0 ^ 2 ^ 1 ) (they all belong to S z ) describe a good

strategy if and only if this is true:

For each z ) j for which 7 * does not assume its minimum
(in j a ), we have pj = 0. 4

specifically we mean the middle terms — 6 pJ* and bp[* in (19:9:a) and (19 :9:c).
1 These terms correspond to 81 = s 2 , in our present set-up to z\ = z iy and since the
z 1 , z% are continuous variables, the probability of their (fortuitous) coincidence is indeed 0.

Mathematically one may describe these operations by saying that we are now
carrying out the limiting process S -* « .

3 We mean in j and not in s, j\

4 The formulae (19:7*), (19:9:a*), (19:9:6*), (19:9:c*) and this criterion could also have

— ► — ►

been derived directly by discussing this n continuous 1 ' arrangement, with the p % p f i

in place of the £ , 17 from the start. We preferred the lengthier and more explicit proce-
dure followed in 19.4.-19.7. in order to make the rigor and the completeness of our proce-
dure apparent. The reader will find it a good exercise to carry out the shorter direct
discussion, mentioned above.

It would be tempting to build up a theory of games, into which such continuous
parameters enter, systematically and directly; i.e. in sufficient generality for applications
like the present one, and without the necessity for a limiting process from discrete games.

An interesting step in this direction was taken by J. Ville in the work referred to in
footnote 1 on p. 164: pp. 110-113 loc. cit. The continuity assumptions made there seem,
however, to be too restrictive for many applications, — in particular for the present one.

POKER AND BLUFFING

199

19.8. Mathematical Determination of the Solution

19.8.1. We now proceed to the determination of the good strategies p *,
i.e. of the solution of the implicit condition (19:B) of 19.7.

Assume first that p| > 0 ever happens. 1 For such a z necessarily
Min, 7 J = y\ hence 7* ^ y\ i.e.

7*2 - 7! ^ 0.

Substituting (19:9:a*), (19:9:b*) into this gives

(19:11) (a — b) p z 2 'dzi — f * pl*dzi) + 2 b f\‘ z 'dzi g 0.

Now let z° be the upper limit of these z with p\ > 0. 2 Then (19:11) holds
by continuity for z = z° too. As p\* > 0 does not occur for z 1 > z ° — by

hypothesis — so the f p|» dzi term in (19:11) is now 0. So we may write it

with + instead of — , and (19:11) becomes:

(a — b) p l 2 'dzi + 26 J p\^dz\ ^ 0.

But p{> is always ^ 0 and sometimes > 0, by hypothesis; hence the first
term is > 0. 8 ’ 4 The second term is clearly ^ 0. So we have derived
a contradiction. I.e. we have shown

(19:12) p\ m 0. 6

19.8.2. Having eliminated j = 2 we now analyze the relationship of
j = 1 and j = 3. Since p\ — 0 so p\ + p\ = 1 i.e. :

(19:13) Pi = 1 - Pi,

and consequently

(19:14) 0 £ pi S 1.

Now there may exist in the interval 0 ^ z ^ 1 subintervals in which
always p* = 0 or always p* = 1.® A z which is not inside any interval of

1 I.e. that the good strategy under consideration provides for j — 2, i.e. “low”
bidding with (the intention of) subsequent “Seeing,” under certain conditions.

* I.e. the greatest 2°for which p{ > 0 occurs arbitrarily near to z°. (But we do not
require pj > 0 for all 2 < z°.) This z° exists certainly if the z with pj > 0 exist.

* Of course a — b > 0.

4 It does not seem necessary to go into the detailed fine points of the theory of integra-
tion, measure, etc. We assume that our functions are smooth enough so that a positive
function has a positive integral etc. An exact treatment could be given with ease if we
made use of the pertinent mathematical theories mentioned above.

•The reader should reformulate this verbally: We excluded “low” bids with (the
intention of) subsequent “Seeing” by analysing conditions for the (hypothetical) upper
limit of the hands for which this would be done; and showed that near there, at least, an
outright “high” bid would be preferable.

This is, of course, conditioned by our simplification which forbids “overbidding.”
6 I.e. where the strategy directs the player to bid always “high,” or where it directs
him to bid always “low” (with subsequent “Passing”).

200

ZERO-SUM TWO-PERSON GAMES: EXAMPLES

either kind — i.e. arbitrarily near to which both p*' ^ 0 and pf y* 1 occur —
will be called intermediate. Since p*' ^ 0 or p*' ^ 1 (i.e. pj' ^ 0) imply
Min, y‘ f = 7*' or 75' respectively, therefore we see: Both y{ ^ 73' and
7*/ ;> 7j' occur arbitrarily near to an intermediate z. Hence for such a z,
y\ = y\ by continuity, 1 i.e.

75 ~ 7*i = 0.

Substituting (19:9:a*), (19:9:c*) and recalling (19:12), (19:13), gives

(a + b) j* p\'dzi — (a — b) p\'dzi + 26 J (1 — p\i)dz\ = 0
i.e.

(19:15) (a + 6 ) ([’ P\'dz 1 - Pl'rfz.) + 26(1 - 2) = 0.

Consider next two intermediate z f , z". Apply (19:15) to z = z' and
2 = z" and subtract. Then

2(a + 6) p*,.d2, - 26(2” - 2') = 0

obtains, i.e.

r i n

(19:16) ^ | ^

Verbally: Between two intermediate z', z" the average of p{ is -y
So neither p\ = 0 nor p\ = 1 can hold throughout the interval

z' z ^ 2"

since that would yield the average 0 or 1. Hence this interval must contain
(at least) a further intermediate z, i.e. between any two intermediate places
there lies (at least) a third one. Iteration of this result shows that between
two intermediate z', z" the further intermediate z lie everywhere dense.
Hence the z', z" for which (19:16) holds lie everywhere dense between z', z".
But then (19:16) must hold for all z', z" between z', z", by continuity. 2

This leaves no alternative but that p* = - ^ everywhere between z', z". s

1 The 7 * are defined by integrals (19:9 :a*), (19:9:b*), ( 19 :9:c*), hence they are certainly
continuous.

* The integral in (19:16) is certainly continuous.

* Clearly isolated exceptions covering a z area 0 — i.e. of total probability zero (e.g. a
finite number of fixed z's) — could be permitted. They alter no integrals. An exact
mathematical treatment would be easy but does not seem to be called for in this context

(cf. footnote 4 on p. 199). So it seems simplest to assume p[ — — 7-7 in 2' & z £ 2"

a -+* o

without any exceptions.

This ought to be kept in mind when appraising the formulae of the next pages which
deal with the interval 2’ 2 * z £ 2" on one hand, and with the intervals 0 £ z < V and
l” < z' g 1 on the other; i.e. which count the points 2\ 2" to the first-mentioned interval.
This is, of course, irrelevant: two fixed isolated points — 2* and 2” in this case — could
be disposed of in any way (cf. above).

The reader must observe, however, that while there is no significant difference

POKER AND BLUFFING

201

19.8.3. Now if intermediate z exist at all, then there exists a smallest
one and a largest one; choose 2 ', 2 " as these. We have

(19:17) pi = a ^_ ' ^ throughout 2 ' ^ z ^ 2 ".

If no intermediate z exist, then we must have p\ = 0 (for all z) or p\ ss 1
(for all z). It is easy to see that neither is a solution . 1 Thus intermediate z
do exist and with them 2 ', 2 " exist and (19:17) is valid.

19.8.4. The left hand side of (19:15) is 7 } — y\ for all z; hence for z = 1

y\ - y\ = (a + b) f 1 p\^dzi > 0,

Jo

(since = 0 is excluded). By continuity y\ — y\ > 0, i.e. 7 * < y \ remains
true even when z is merely near enough to 1 . Hence p\ = 0, i.e. p* = 1 for
these z. Thus (19:17) necessitates 2 " < 1. Now no intermediate z exists
in 2 " z ^ 1 ; hence we have p\ = 0 or p\ = 1 throughout this interval.
Our preceding result excludes the former. Hence

(19:18) p\ = 1 throughout 2 " ^ z ^ 1.

19.8.5. Consider finally the lower end of (19:17), 2 '. If 2 ' > 0 then we
have an interval 0 g z ^ 2 '. This interval contains no intermediate z;
hence we have p* = 0 or p\ = 1 throughout 0 ^ z g 2 '. The first derivative
of y\ — y\, i.e. of the left side of (19:15), is clearly 2(a + b)p\ — 2b. Hence
in 0 ^ z < 2 ' this derivative is 2 (a + b) • 0 — 2b = —2 b < 0 if p\ s 0
there, 2 (a + 6 ) • 1 — 2b = 2 a > 0 if p\ = 1 there, i.e. y\ — y\ is monotone
decreasing or increasing respectively, throughout 0 ^ z < 2 '. Since its
value is 0 at the upper end (the intermediate point 2 '), we have 73 — y\ > 0
or < 0 respectively, i.e. 7* < y\ or y\ < y\ respectively, throughout
0 ^ z < 2 '. The former necessitates p\ s 0, p* s 1 the latter p^ = 0 in
0 ^ 2 < 2 '; but the hypotheses with which we started were p\ ss 0 or pi = 1
respectively, there. So there is u contradiction in each case.

Consequently

(19:19) 2 ' = 0.

19.8.6. And now we determine 2 " by expressing the validity of (19:15)
for the intermediate z = 2 ' = 0. Then (19:15) becomes

— (a -f- b) J p\'dzi -t- 26 = 0

J* pldzi =

between a < and a «£ when the z } s themselves are compared, this is not so for the 7 ;.
Thus we saw that y[ > y\ implies pj = 0, while y\ £ y\ has no such consequence.
(Cf. also the discussion of Fig. 41 and of Figs. 47, 48.)

1 I.e. bidding “low” (with subsequent “Pass") under all conditions is not a good
strategy; nor is bidding “high” under all conditions.

Mathematical proof: For p\ hO: Compute 7 ? • —&, 7 i “ & hence 7® < 7J
contradicting p\ - 1 ^ 0. For p{ m 1: Compute 7 ? ■ 7i “ & hence 7 } < 7 ?
contradicting p® - 1 ^ 0 .

202 ZERO-SUM TWO-PERSON GAMES: EXAMPLES
But (19:17), (19:18), (19:19) give

- 2") • 1

= 1 - “ . • 2".
a + b

So we have

i a zn _ 26

o -(- 6 a + 6

a *>i _ i 2b

a + 6 a “b 6

a — 6
o + 6’

i.e.

(19:20)

i" = a ~ b
a

Combining (19:17), (19:18), (19:19), (19:20) gives:

(19:21)

Pi

a -f- b
= 1

for

for

0 g z g

a — b

<z ^ 1.

Together with (19:12), (19:13) this characterizes the strategy completely.

19.9. Detailed Analysis of the Solution

19.9.1. The results of 19.8. ascertain that there exists one and only one
good strategy in the form of Poker under consideration. 1 It is described
by (19:21), (19:12), (19:13) in 19.8. We shall give a graphical picture of this
strategy which will make it easier to discuss it verbally in what follows.
(Cf. Figure 40. The actual proportions of this figure correspond to
a/b ~ 3.)

The line plots the curve p = p\. Thus the height of above

the line p = 0 is the probability of a “high” bid: p\; the height of the line
p = 1 above is the probability of a “low” bid (necessarily with sub-

sequent “pass”): p| = 1 — p*.

19.9.2. The formulae (19:9:a*), (19:9:b*), (19 :9:c*) of 19.7. permit us
now to compute the coefficients y]. We give the graphical representations
instead of the formulae, leaving the elementary verification to the reader.
(Cf. Figure 41. The actual proportions are those of Figure 40, i.e. a/b ~ 3 —

cf. there.) The line plots the curve 7 = 7!; the line plots the

curve 7 = 7j ; the line plots the curve y = y\. The figure shows that

1 We have actually shown only that nothing else but the strategy determined in 19.8.
can be good. That this strategy is indeed good, could be concluded from the established
existence of (at least) a good strategy, although our passage to the “continuous” case
may there create some doubt. But we shall verify below that the strategy in question is
good, i.e. that it fulfills (19:B) of 19.7.

POKER AND BLUFFING

203

and (i.e. y\ and 7$) coincide in 0 ^ and that and

a

(i.e. 75 and 7J) coincide in £ z £ 1 . All three curves are made

CL

P

1

b

1

l

1

1

1

i

1

i

1

l

1

1

1

I

a + b

i

i

i

i

1

0 a - b 1

a

Figure 40.

7* at the critical points z = 0, - 1 are given in the figure. 1

CL

1 The simple computational verification of these results is left to the reader.

204 ZERO-SUM TWO-PERSON GAMES: EXAMPLES

19 . 9 . 3 . Comparison of Figures 40 and 41 shows that our strategy is

indeed good, i.e. that it fulfills (19 :B) of 19.7. Indeed: In 0 Sjj z £

where both p[ ^ 0, pf ^ 0 both and 75 are the lowest curves, i.e. equal
to Min, 7'. In < z ^ 1 where only p\ 0 there only 7; is the

CL

lowest curve, i.e. equal to Min, yj. (The behavior of 75 does not matter,
since always p\ = 0.)

We can also compute K from (19:7*) in 19.7., the value of a play. K = 0
is easily obtained; and this is the value to be expected since the game is
symmetric.

19.10. Interpretation of the Solution

19 . 10 . 1 . The results of 19.8., 19.9., although mathematically complete,
call for a certain amount of verbal comment and interpretation, which we
now proceed to give.

First the picture of the good strategy, as given in Figure 40, indicates
that for a sufficiently strong hand p[ = 1; i.e. that the player should then

bid “high,” and nothing else. This is the case for hands z > ~—~ m For

weaker hands, however, p\ — —rrV Pa = 1 — p\ = ~~TTV 80 both pf, pj 0;
i.e. the player should then bid irregularly “high” and “low” (with specified
probabilities). This is the case for hands z ^ -• The “high” bids

(in this case) should be rarer than the “low” ones, indeed ~ and a > b.

Pi b

This last formula shows too that the last kind of “high” bids become
increasingly rare if the cost of a “high” bid (relative to a “low” one)
increases.

Now these “high” bids on “weak” hands — made irregularly, governed
by (specified) probabilities only, and getting rarer when the cost of “high”
bidding is increased — invite an obvious interpretation: These are the
“Bluffs” of ordinary Poker.

Due to the extreme simplifications which we applied to Poker for the
purpose of this discussion, “Bluffing” comes up in a very rudimentary form
only; but the symptoms are nevertheless unmistakable: The player is

advised to bid always “high” on a strong hand (z > and to bid

mostly “low” ^with the probability - on a “low” one ^

but with occasional, irregularly distributed “Bluffs” (with the probability

_lA

a + b)

POKER AND BLUFFING

265

19 . 10 . 2 . Second, the conditions in the zone of “ Bluffing,” 0 £ z <£ - —

a

throw some light on another matter too, — the consequences of deviating
from the good strategy, “ permanent optimality,” “ defensive,” “ offensive,”
as discussed in 17.10.1., 17.10.2.

Assume that player 2 deviates from the good strategy, i.e. uses proba-
bilities a) which may differ from the p) obtained above. Assume, further-
more, that player 1 still uses those p‘, i.e. the good strategy. Then we can
use for the y) of (19:9a*), (19:9:b*), (19:9:c*) in 19.7., the graphical repre-
sentation of Figure 41, and express the outcome of the play — for player 1 —
by (19:7*) in 19.7.

(19:22) K - 5) j 1 ytfdz.

i

Consequently player 2’s a J are optimal against player Us p* if the analogue
of the condition (19:8) in 19.6. is fulfilled:

(19 :C) For each pair z, j for which 7* does not assume its minimum

(in j 0 we have <r* = 0.

I.e. (19 :C) is necessary and sufficient for a * being just as good against pj as pj
itself, — that is, giving a K = 0. Otherwise is worse, — that is, giving a
K > 0. In other words:

(19 :D) A mistake , i.e. a strategy a] which deviates from the good

strategy p ) will cause no losses when the opponent sticks to the
good strategy if and only if the a * fulfill (19 :C) above.

Now one glance at Figure 41 suffices to make it clear that (19 :C) means

or* = <r* ss 0 for z > ^t merely g\ = 0 for z ^ *~~ ' 9 I- e - : (19:C)

prescribes a high” bidding, and nothing else, for strong hands >

it forbids “low” bidding with subsequent “Seeing” for all hands, but it
foils to prescribe the probability ratio of “high” bidding and of “low”
bidding (with subsequent “Passing”) for weak hands, i.e. in the zone of

“Bluffing” (2 £ a -^

19 . 10 . 3 . Thus any deviation from the good strategy which involves
more than just incorrect “Bluffing,” leads to immediate losses. It suffices
for the opponent to stick to the good strategy. Incorrect “Bluffing”
causes no losses against an opponent playing the good strategy; but the

1 We mean in j % and not in j\

a — b

1 Actually even <n 7* 0 would be permitted at the one place z — — - — But this

isolated value of z has probability 0 and so it can be disregarded. Cf. footnote 3 on

p. 200.

206

ZERO-SUM TWO-PERSON GAMES: EXAMPLES

opponent could inflict losses by deviating appropriately from the good
strategy. I.e. the importance of “Bluffing” lies not in the actual play,
played against a good player, but in the protection which it provides against
the opponent’s potential deviations from the good strategy. This is in
agreement with the remarks made at the end of 19.2., particularly with the
second interpretation which we proposed there for “ Bluffing.” 1 Indeed, the
element of uncertainty created by “Bluffing” is just that type of constraint
on the opponent’s strategy to which we referred there, and which was ana-
lyzed at the end of 19.2.

Our results on “bluffing” fit in also with the conclusions of 17.10.2.
We see that the unique good strategy of this variant of Poker is not per-
manently optimal; hence no permanently optimal strategy exists there.
(Cf. the first remarks in 17.10.2., particularly footnote 3 on p. 163.) And
“Bluffing” is a defensive measure in the sense discussed in the second half
of 17.10.2.

19 . 10 . 4 . Third and last, let us take a look at the offensive steps indicated
loc. cit., i.e. the deviations from good strategy by which a player can profit
from his opponent’s failure to “Bluff” correctly.

We reverse the roles: Let player 1 “ Bluff” incorrectly, i.e. use p* different
from those of Figure 40. Since only incorrect “Bluffing” is involved, we
still assume

p* 2 = 0 for all z

p \ ~ * } for all z > -•

Pa = Of <*

So we are interested only in the consequences of

(19:23)

Pi ^

for some

. a — b 2
z = z o < 2

a + b a

The left hand side of (19:15) in 19.8. is still a valid expression for y\ — y\.

Consider now a z < z 0 . Then ^ in (19:23) leaves / p*idzi unaffected, but

Jo

it * ncreases f 1 p *idzi hence it ^ ecreases the left hand side of (19:15), i.e.
decreases J z increases

7a “ 7*- Since y\ — y\ would be 0 without the change (19:23) (cf. Figure
41), so it will now be ^ 0. I.e. y\ ^ y\. Consider next a z in

^ ^ a — b

z 0 < z £

1 All this holds for the form of Poker now under consideration. For further view-
points cf. 19.16.

* We need this really for more than one 2 , cf. footnote 3 on p. 200. The simplest
assumption is that these inequalities hold in a small neighborhood of the z 0 in question.

It would be easy to treat this matter rigorously in the sense of footnote 4 on p. 199
and of footnote 3 on p. 200. We refrain from doing it for the reason stated there.

POKER AND BLUFFING

207

Then ^ in (19:23) decreases L ** leaves J p{4zi unaffected;

increases

hence it decreases t^e left hand side of (19:15), i.e. y\ — y\. Since y\ — >5

would be 0 without the change (19:23) (cf. Fig. 41), so it will now be ^ 0.
I.e. y\ ^ 7 *. Summing up:

(19 :E) The change (19:23) with ^ causes

for z < z o,

o — 6

7s 5 y\

7i ^ 7'

for

Zo < z ^

Hence the opponent can gain, i.e. decrease the K of (19:22), by using <rj
which differ from the present pj: For z < z 0 by increasing at the expense

°i

of i.e. by f ecreas | n g a z f rom ^ e value of p{, — ^-r> to the extreme value *?•
increasing a + b 1

And for z 0 < z ^ ~ by increasing ^ at the expense of i.e. by ^ecreairing

<r{ from the value of p{, — r to the extreme value In other words:

1 a + b 0

(19:F)

If the opponent “Bluffs” too

much

little

for a certain hand z 0 , then

he can be punished by the following deviations from the good
less

strategy: “Bluffing” for hands weaker than z 0 , and “Bluff-
more

ing” for hands stronger than z 0 .

I.e. by imitating his mistake for hands which are stronger
than z 0 and by doing the opposite for weaker ones.

These are the precise details of how correct “Bluffing” protects against
too much or too little “Bluffing” of the opponent, and its immediate
consequences. Reflections in this direction could be carried even beyond
this point, but we do not propose to pursue this subject any further.

19.11. More Genera) Forms of Poker

19 . 11 . While the discussions which we have now concluded throw a
good deal of light on the strategical structure and the possibilities of Poker,
they succeeded only due to our far reaching simplification of the rules of
the game. These simplifications were formulated and imposed in 19.1.,
19.3. and 19.7. For a real understanding of the game we should now
make an effort to remove them.

By this we do not mean that all the fanciful complications of the game
which we have eliminated (cf. 19.1.) must necessarily be reinstated, 1

1 Nor do we wish, yet to consider anything but a two-person game!

208

ZERO-SUM TWO-PERSON GAMES: EXAMPLES

but some simple and important features of the game were equally lost and
their reconsideration would be of great advantage. We mean in particular:

(A) The “hands” should be discrete, and not continuous. (Cf. 19.7.)

(B) There should be more than two possible ways to bid. (Cf. 19.3.)

(C) There should be more than one opportunity for each player to bid,
and alternating bids, instead of simultaneous ones, should also be considered.
(Cf. 19.3.)

The problem of meeting these desiderata (A), (B), (C) simultaneously —
and finding the good strategies — is unsolved. Therefore we must be satis-
fied for the moment to add (A), (B), (C) separately.

The complete solutions for (A) and for (B) are known, while for (C)
only a very limited amount of progress has been made. It would lead
too far to give all these mathematical deductions in detail, but we shall
report briefly the results concerning (A), (B), (C).

19.12. Discrete Hands

19.12.1. Consider first (A). I.e. let us return to the discrete scale of
hands s = 1, • • • , S as introduced at the end of 19.1.2., and used in
19.4-19.7. In this case the solution is in many ways similar to that of
Figure 40. Generally pj = 0 and there exists a certain s° such that p{ = 1
for s > s°, while p} 5* 0, 1 for s < s # . Also, if we change to the z scale (cf.
^0 2 d b

Fig. 39) , then -5 r is very nearly — — • 1 So we have a zone of “ Bluffing ”

o — I a

and above it a zone of “high” bids, — just as in Fig. 40.

But the Pi for s < $°, i.e. in the zone of “Bluffing,” are not at all equal

to or near to the ^ of Fig. 40. 2 They oscillate around this value by

amounts which depend on certain arithmetical peculiarities of S but do not
tend to disappear for S — > <*> . The averages of the p\ however, tend to

— r- 3 In other words:
a + b

The good strategy of the discrete game is very much like the good
strategy of the continuous game: this is true for all details as far as the
division into two zones (of “Bluffing” and of “high” bids) is concerned;
also for the positions and sizes of these zones, and for the events in the zone
of “high” bids. But in the zone of “Bluffing” it applies only to average
statements (concerning several hands of approximately equal strength).
The precise procedures for individual hands may differ widely from those

1 Precisely: — ► - — - for S — ► « .

o — 1 a

b

8 I.e. not pf — ► ^ for S — ► oo whatever the variability of s.

1 b

8 Actually ^ (pf + p} +1 ) - " ^ ^ for most a < «°.

POKER AND BLUFFING 209

given in Figure 40, and depend on arithmetical peculiarities of s and S
(with respect to a/b). 1 * *

19.12.2. Thus the strategy which corresponds more precisely to Figure 40

— i.e. where p[ « — ^ for all « < 8° — is not good, and it differs quite

considerably from the good one. Nevertheless it can be shown that' the
maximal loss which can be incurred by playing this “average” strategy
is not great. More precisely, it tends to 0 for S — » » .*

So we see: In the discrete game the correct way of “Bluffing” has a
very complicated “fine structure,” which however secures only an extremely
small advantage to the player who uses it.

This phenomenon is possibly typical, and recurs in much more compli-
cated real games. It shows how extremely careful one must be in asserting
or expecting continuity in this theory.’ But the practical importance
— i.e. the gains and losses caused — seems to be small, and the whole thing
is probably terra incognita to even the most experienced players.

19.13. m possible Bids

19.13.1. Consider, second, (B) : I.e. let us keep the hands continuous, but
permit bidding in more than two ways. I.e. we replace the two bids

a > b ( > 0)

by a greater number, say m ) ordered :

ai > a 2 > • • • > a m _i > a m (> 0).

In this case too the solution bears a certain similarity to that of Figure 40. 4
There exists a certain z° 5 such that for z > z° the player should make the
highest bid, and nothing else, while for z < z° he should make irregularly
various bids (always including the highest bid o, but also others), with
specified probabilities. Which bids he must make and with what proba-

1 Thus in the equivalent of Figure 40, the left part of the figure will not be a straight
line - ~ k in 0 ^ 2 ^ but one which oscillates violently around this

average.

* It is actually of the order 1/S. Remember that in real Poker S is about 2\ millions.
(Cf. footnote 4 on p. 187.)

1 Recall in this connection the remarks made in the second part of footnote 4 on p.
198.

4 It has actually been determined only under the further restriction of the rules which
forbids “Seeing” a higher bid. I.e. each player is expected to make his final, highest bid
at once, and to “Pass” (and accept the consequences) if the opponent's bid should turn
out higher than his.

210

ZERO-SUM TWO-PERSON GAMES: EXAMPLES

bilities, is determined by the value of z. 1 So we have a zone of “Bluffing”
and above it a zone of “high” bids — actually of the highest bid and nothing
else — just as in Figure 40. But the “Bluffing” — in its own zone z S z° —
has a much more complicated and varying structure than in Figure 40.

We shall not go into a detailed analysis of this structure, although
it offers some quite interesting aspects. We shall, however, mention one of
its peculiarities.

19.13.2. Let two values

a > b > 0

be given, and use them as highest and lowest bids:

CL l = CL) CL m — b.

Now let m — > oo and choose the remaining bids a 2 , * • • , a m _i so that they
fill the interval

(19:24) b

with unlimited increasing density. (Cf. the two examples to be given in
footnote 2 below.) If the good strategy described above now tends to a
limit — i.e. to an asymptotic strategy for m — » oc — then one could interpret
this as a good strategy for the game in which only upper and lower bounds
are set for the bids (a and b), and the bids can be anything in between (i.e.
in (19:24)). I.e. the requirement of a minimum interval between bids
mentioned at the beginning of 19.3. is removed.

Now this is not the case. E.g. we can interpolate the a 2 , • • • , a m _i
between a x = a and a m = b both in arithmetic and in geometric sequence. 2
In both cases an asymptotic strategy obtains for m — > » but the two strate-
gies differ in many essential details.

If we consider the game in which all bids (19 :24) are permitted, as one
in its own right, then a direct determination of its good strategies is possible.

1 If the bids which he must make are ai, a Pl a q , • • • , a„(l < p < q < • • < n),

then it can be shown that their probabilities must be

111 1 ( 1 , 1 , , 1 \

— i — , — > . . . — i 1 c 1 — -h * • • H J

coi ca P ca q ca n V oi a p

respectively. I.e. if a certain bid is to be made at all, then its probability must be
inversely proportional to the cost.

Which o p , a q , • • • a m actually occur for a given z is determined by a more com-
plicated criterion, which we shall not discuss here.

Observe that the c above was needed only to make the sum of all probabilities
equal to 1. The reader may verify for himself that the probabilities in Figure 40 have
the above values.

* The first one is defined by

a f “ ^ "x (( m ~ V) a + (V “ 1 )b) for p - 1, 2, • •• , m - 1, m
the second one is defined by

Op — m ~\/ a m ~ p b p ~ l for p — 1, 2, • • • , m — 1, m.

POKER AND BLUFFING 211

It turns out that both strategies mentioned above are good, together with
many others.

This chows to what complications the abandonment of a minimum inter-
val between bids can lead: a good strategy of the limiting case cannot be an
approximation for the good strategies of all nearby cases with a finite num-
ber of bids. The concluding remarks of 19.12. are thus re-emphasized.

10.14. Alternate Bidding

19 . 14 . 1 . Third and last, consider (C) : The only progress so far made in
this direction is that we can replace the simultaneous bids of the two players
by two successive ones; i.e. by an arrangement in which player 1 bids first
and player 2 bids afterwards.

Thus the rules stated in 19.4. are modified as follows:

First each player obtains, by a chance move, his hand 8 = 1, • • • , S,
each one of these numbers having the same probability l/S. We denote
the hands of players 1, 2, by s h s 2 respectively.

After this 1 player 1 will, by a personal move, choose either a or 6, — the
“high” or the “low” bid. 2 He does this informed about his own hand
but not about the opponent's hand. If his bid is “low,” then the play is
concluded. If his bid is “high,” then player 2 will, by a personal move,
choose either a or 6, — the “high ” or the “low ” bid. 3 He does this informed
about his own hand, and about the opponent’s choice, but not his hand.

This is the play. When it is concluded, the payments are made as

>

follows: If player 1 bids “low,” then for s i = s 2 player 1 obtains from player

<

b >

2 the amount 0 respectively. If both players bid “ high,” then for s i = s 2
-b <

a

player 1 obtains from player 2 the amount 0 respectively. If player 1

— a

bids “high” and player 2 bids “low,” then player 1 obtains from player 2
the amount b. 4

19 . 14 . 2 . The discussion of the pure and mixed strategies can now be
carried out, essentially as we did for our original variant of Poker in 19.5.

We give the main lines of this discussion in a way which will be per-
fectly clear for the reader who remembers the procedure of 19.4.-19.7.

A pure strategy in this game consists clearly of the following specifica-
tions: to state for every hand s = 1, • • • , S whether a “high” or a “low”
bid will be made. It is simpler to describe this by a numerical index
i 9 = 1, 2; i, = 1 meaning a “high” bid, i, = 2 meaning a “low” bid. Thus

1 We continue from here on as if player 2 had already made the “low” bid, and this
were player l’s turn to “See” or to “Overbid.” We disregard “Passing” at this stage.

* I.e. “Overbid” or “See,” cf. footnote 1 above.

* I.e. “See” or “Pass.” Observe the shift of meaning since footnote 2 above.

4 In interpreting these rules, recall the above footnotes. From the formalistic point
of view, footnote 1 on p. 191 should be recalled, mutatis mutandis.

212

ZERO-SUM TWO-PERSON GAMES: EXAMPLES

the strategy is a specification of such an index t, for every 8 = 1, • • • , S
i.e. of a sequence ii, ••• , i s .

This applies to both players 1 and 2; accordingly we shall denote the
above strategy by Xi(i h • • • , is) or • • • , is). Thus each player

has the same number of strategies, — as many as there are sequences
t‘i, • • • , is ; i.e. precisely 2 5 . With the notations of 11.2.2.

Pi = Pi - P = 2 s .

(But the game is not symmetrical !)

We must now express the payment which player 1 receives if the strate-
gies • • • , is), 2t(j i, • • • , js) are used by the two players. This is

the matrix element • • • , ia\ji, * * • , js ). If the players have

actually hands 8 h then the payment received by player 1 can be expressed
in this way (using the rules stated above): It is JB, an (.-« t) (i #i , j$) where
sgn(si — is the sign of 8i — 8% and where the three functions

<£+(». j), <Co(*, j), j)

can be represented by the following matrix schemes:

\j

t \

1

2

1

—a

b

2

-6

-b

\j

i \

1

2

1

0

b

2

0

0

\j

t \

1

2

1

a

&

2

b

6

Figure 42. Figure 43. Figure 44.

Now 8i t s 2 originate from chance moves, as described above. Hence:

1

3 C(il> 1 is\j If f js) = £2 X

19 . 14 . 3 . We now pass to the mixed strategies in the sense of 17.2. These
— ► — >

are vectors £ , tj belonging to Sp. We must index the components of these
vectors like the (pure) strategies: we must write £< t i a , m, i a instead

Of (r t f Vr f *

We express (17 :2) of 17.4.1. which evaluates the expectation of player Us
gain

K( £ , n ) — ^ 3C(*1, , ia\j if * * • fjs)ii x i a Vi t i a

*'i. * • • . isj v

= 52 X <,W» V

•1. • • • t is. A. • •• fj ( t t

POKER AND BLUFFING

213

There is an advantage in interchanging the two S and writing

K( £ , V ) = gi (Lj, i» s ) £>,

• p*j »f. ‘ * ’ » ' ’ * - J a

If we now put
(19:25)

(19:26)

£♦ »,»

*!.••• excluding t*

w. v

;s excluding

then the above equation becomes

(19:27) K( £ , ^«^n(# l -# i )(^,

19 . 14 . 4 . All this is precisely as in 19.5.2. As there, (19:25) shows that

— ►

p'i is the probability that player 1, using the mixed strategy £ will choose i
when his hand is «j. (19:26) shows that a}* is the probability that player 2,

using the mixed strategy ij will choose j when his hand is s 2 . It is again

— * — *

clear intuitively that the expectation value K( £ , rj ) depends upon these

probabilities only, and not on the underlying probabilities £ <t ig , t ;, jt .

themselves. (19:27) expresses this and could have easily been derived
directly, on this basis.

It is also clear, both from the meaning of the p*», cr*« and from their formal
definitions (19:25), (19:26), that they fulfill the conditions :

(19:28)

6 0 X pi ' "

(19:29)

t). ^ o £*;«= h

and that any p*>, cr*> which fulfill these conditions can be obtained from

suitable £ , ij by (19:25), (19:26). (Cf. the corresponding step in 19.5.?.
particularly footnote 1 on p. 194.) It is therefore opportune to form the
2-dimensional vectors

P *■ = {p*‘, P{*}, <7 ** = l<7{*,

— ► -4

Then (19:28), (19:29) state precisely that all p •>, a *» belong to St.

214

ZERO-SUM TWO-PERSON GAMES: EXAMPLES

Thus { (or ij ) was a vector in S B i.e. depending on 0 — 1 = 2 s — 1

constants; the p •» (or a •*) are S vectors in S 2 i.e. each one depends on one
numerical constant, hence they amount together to S numerical constants.
So we have reduced 2 s — 1 to S. (Cf. the end of 19.5.3.)

19 . 14 . 5 . We now rewrite (19:27) as in 19.6.

(19:30) K(7‘, • • ' , 7 S |7‘, ■■■ ,7 s )

with the coefficients

S 2

2

)(h

8 V t

i.e. using the matrix schemes of Figures 42-44,

«,-i s

(19:31 :a) 7 * 1 * = -§ | ^ (-ap f i l - &P 2 O + ^ (api 1 + 6p* 2 o|

*i»i *,-#,+ 1

« a -i s

(19:31 :b) 7?’ = -g | ^ (*>p'‘ - £>p* 2‘) + Vi’ + ^ (i>P*‘ + bpioj'

#1 * 1 *1 "* ** 1

Since the game is no longer symmetric, we need also the corresponding
formulae in which the roles of the two players are interchanged. This is:

(19:32) K( p P s \ a l , • • • , <r s ) = g ^ $I‘P,\

#1,1

with the coefficients

g * ^2 ^ <C#»n (#,-#,) (t. j)(?)'

hJ

i.e. using the matrix schemes of Figures 42-44,

•i-l s

(19:33:a) $*» = ^ ^ (a*! 1 + bai*) + ^ {-aa\* + fccrj*) |

« j-i #,-#,+1

#t-l 8

(19:33 :b) i}‘ = g | ^ (M* + ^5’) + ^ (—&»{■ —

■■ 1 #t " »j 4- 1

The criteria for good strategies are now essentially repetitions of those in
19.6. I.e. due to the asymmetry of the variant now under consideration
our present criterion will be obtained from the general criterion (17:D)

POKER AND BLUFFING

215

of 17.9. in the same way as that of 19.6. could be obtained from the sym-
metrical criterion at the end of 17.11.2. I.e.:

(19:G) The p l , • • , p 8 and the a l , • • • , a 8 — they all belong

to St — describe good strategies# and only if this is true:

For each * s , j, for which 7}* does not assume its minimum
(in j *) we have cr** = 0. For each s 1 , i for which does not
assume its maximum (in i *) we have pj» = 0.

19.14.6. Now we replace the discrete hands $1, 8 % by continuous ones,
in the sense of 19.7. (Cf. in particular Figure 39 there.) As described in

19.7. this replaces the vectors p •», a •* («i, 8% = 1, • • • , S) by vectors

p *», <r •* (0 2* z 1, *i^l), which are still probability vectors of the same nature
as before, i.e. belonging to S*. So the components p}», <r** make place for the
components p*», <rj*. Similarly the $*», 7}* become 75*. The sums in our
formulae (19:30), (19:31:a), (19:31:b), and (19:32), (19:33:a), (19:33:b) go
over into integrals, just as in (19:7*), (19:9 :a*), (19:9:b*), (19:9:c*) in 19.7.
So we obtain:

(19:30*) K = S / 0 7jvj<bi»

;

(19:31 :a*) y\' — [ ' ( — ap' 1 — bp\>)dzi + f (ap^ + bp}i)dz h

(19:31 :b*) y\' = ' (bp\> — bp‘ t >)dzi + J* (bp\i + bptfdzi,

and

(19:32*) K = J) fj «*i pj-dz,,

(19:33:a*) 4*> = / * (cur? + bff‘ t *)dz t + f ( — an** + batfdzt,

(19:33:b*) (?w*> + bo\<)dzi + f (-iw*» — ba\i)dz%.

JO J *j

Our criterion for good strategies is now equally transformed. (This is the
same transition as that from the discrete criterion of 19.6. to the continuous
criterion of 19.7.) We obtain

(19:H) The p *i and the <r '• —(0 g z u z t g 1) they all belong to St—

describe good strategies if and only if this is true:

For each Zt, j for which >*■ does not assume its minimum
(in j *) we have <r*« = 0. For each t x , i for which 4*i does not
assume its maximum (in i ! ) we have p*> = 0.

1 We mean in j (*) and not in »«, j (*i, *) !

* We mean in j (») and not in t t , j (*i, j) 1

216

ZERO-SUM TWO-PERSON GAMES: EXAMPLES

19.15. Mathematical Description of All Solutions

19.16.1. The determination of the good strategies p * and a *, i.e. of the
solutions of the implicit condition stated at the end of 19.14., can be carried
out completely. The mathematical methods which achieve this are similar
to those with which we determined in 19.8. the good strategies of our original
variant of Poker, — i.e. the solutions of the implicit condition stated at the
end of 19.7.

We shall not give the mathematical discussion here, but we shall describe
the good strategies p * and a 9 which it produces.

There exists one and only one good strategy p * while the good strategies

— >

<r 9 form an extensive family. (Cf. Figures 45-46. The actual proportions
of these figures correspond to a/6 ^3.)

a (a -f 36)
g» + 2ab - 6»
a (a -f 36)

The lines plot the curves p = p{ and a = <r\ respectively. Thus the

height of above the line p = 0 (a = 0) is the probability of a “high”

bid, pi (or*) ; the height of the line p = 1 (<r = 1 ) above is the probability

of a “low” bid, p\ = 1 — pj (<r z 2 = 1 — <r{). The irregular
part of the a = a\ curve (in Figure 46) in the interval u ^ z ^ v represents

the multiplicity of the good strategies a z : Indeed, this part of the a = a\
curve is subject to the following (necessary and sufficient) conditions:

I b

\ - a

when Zq =■ u

when u < Zo < v.

Verbally: Between u and v the average of <r\ is 6/a, and on any right end
of this interval the average of a\ is ^ 6/a.

POKER AND BLUFFING

217

Thus both p * and a * exhibit three different types of behavior on these
three intervals: 1

First: 0 £ z < u. Second :u ^ z g v. Third: v <z £ 1. The lengths
of these three intervals are u, v — u, 1 — v, and the somewhat complicated
expressions for u , v can be best remembered with the help of these easily
verified ratios:

u: 1 — v = a — b:a + b
v — u: 1 — v = a:b.

19 . 15 . 2 . The formulae (19:31:a*), (19:31:b*) and (19:33:a*), (19:33:b*)
of 19.14.6. permit us now to compute the coefficients 7 *, SJ. We give (as in
19.9. in Figure 41) the graphical representations, instead of the formulae,

leaving the elementary verification to the reader. For identification of the
— ► — >

P‘, a ‘ as good strategies only the differences, $j — 5J, y\ — y\ matter: Indeed,
the criterion at the end of 19.14. can be formulated as stating that whenever
this difference is > 0 then = 0 or <rf = 0 respectively, and that whenever

lg a = 2a, tg (3 = 26, tg y =» 2 (a - 6)

this difference is < 0 then p{ = 0 or <r[ = 0 respectively. We give therefore
the graphs of these differences. (Cf. Figures 47, 48. The actual propor-
tions are those of Figures 45, 46; i.e. a/b ~ 3, — cf. there.)

The line plots the curve y — yl — yl; the line

plots the curve 5 = 8; — 8J. The irregular part of the 8 = 5* — 8J curve
(in Figure 48) in the interval u z ^ v corresponds to the similarly
irregular part of the <r = <r\ curve (in Figure 46) in the same interval, —

i.e. it also represents the multiplicity of the good strategies <r The restric-
tion to which that part of the a = a\ curve is subjected (cf. the discussion
after Figure 46) means that this part of the 5 = 5* — 5J curve must lie
within the shaded triangle ///////// (cf. Figure 48).

19 . 15 . 3 . Comparison of Figure 45 with Figure 47, and of Figure 46
with Figure 48 shows that our strategies are indeed good, i.e. that they
fulfill (19: H). We leave it to the reader to verify this, in analogy with the
comparison of Figure 40 and Figure 41 in 19.9.

1 Concerning the endpoints of these intervals, etc., cf. footnote 3 on p. 200.

218

ZERO-SUM TWO-PERSON GAMES: EXAMPLES

The value of K can also be obtained from (19:30*) or (19:32*) in 19.14.6.
The result is:

K = bu

(a - b)b * f

cl{cl -f~ 36)

Thus player 1 has a positive expectation value for the play, — i.e. an advan-
tage 2 which is plausibly imputable to his possessing the initiative.

19.16. Interpretation of the Solutions. Conclusions

19 . 16 . 1 . The results of 19.15. should now be discussed in the same way
as those of 19.8., 19.9. were in 19.10. We do not wish to do this at full
length, but just to make a few remarks on this subject.

We see that instead of the two zones of Figure 40 three zones appear
in Figures 45, 46. The highest one (farthest to the right) corresponds to
"high” bids, and nothing else, in all these figures (i.e. for both players).
The behavior of the other zones, however, is not so uniform.

For player 2 (Figure 46) the middle zone describes that kind of “ Bluff-
ing” which we had on the lowest zone in Figure 40, — irregular “high” and
“low” bids on the same hand. But the probabilities, while not entirely
arbitrary, are not uniquely determined as in Figure 40. 3 And there exists
a lowest zone (in Figure 46) where player 2 must always bid “low,” — i.e.
where his hand is too weak for that mixed conduct.

Furthermore, in player 2's middle zone the 7* show the same indifference
as in Figure 41 — 75 — 7* = 0 there, — both in Figure 41 and in Figure 47 —
so the motives for his conduct in this zone are as indirect as those discussed
in the last part of 19.10. Indeed, these “high” bids are more of a defense
against “Bluffing,” than “Bluffing” proper. Since this bid of player 2
concludes the play, there is indeed no motive for the latter, while there is a
need to put a rein on the opponent's “Bluffing” by occasional “high”
bids, — by “Seeing” him.

For player 1 (Figure 45) the situation is different. He must bid “high,”
and nothing else, in the lowest zone; and bid “low,” and nothing else, in
the middle zone. These “high” bids on the very weakest hands — while
the bid on the medium hands is “low” — are aggressive “Bluffing” in its

1 For numerical orientation: If a/6 — 3, which is the ratio on which all our figures are
based, then «■},»»} and K = -•

17

1 For a/b ~ 3 this is about 6/9 (cf. footnote 1 above), i.e. about 11 per cent, of the
“low” bid.

* Cf. the discussion after Figure 46. Indeed, it is even possible to meet those require-
ments with o\ * 0 and 1 only; e.g. <r[ * 0 in the lower fraction and <r\ *■ 1 in the

6

upper - fraction of the middle interval.

The existence of such a solution (i.e. never a \ ^ 0, 1, — by Figure 46 never p[ 9 * 0, 1
either) means, of course, that this variant is strictly determined. But a discussion on
that basis (i.e. with pure strategies) will not disclose solutions like the one actually drawn
in Figure 46.

POKER AND BLUFFING

219

purest form. The are not at all indifferent in this zone of “Bluffing”
(i.e. the lowest zone) : 5* — > 0 there in Figure 48. — i.e. any failure to

“Bluff” under these conditions leads to instant losses.

19.16.2. Summing up: Our new variant of Poker distinguishes two
varieties of “Bluffing”: the purely aggressive one practiced by the player
who has the initiative; and a defensive one — “Seeing” irregularly, even
with a medium hand, the opponent who is suspected of “Bluffing” —
practiced by the player who bids last. Our original variant where the
initiative was split between the two players — because they bid simultane-
ously — contained a procedure which we can now recognize as a mixture of
these two things. 1

All this gives valuable heuristic hints how real Poker — with longer
sequences of (alternating) bids and overbids — ought to be approached.
The mathematical problem is difficult, but probably not beyond the reach of
the techniques that are available. It will be considered in other publications.

1 The variant of E. Borel , referred to in footnote 2 on p. 186, is treated loc. cit. in a
way which bears a certain resemblance to our procedure. Using our terminology, the
course of E. Borel can be described as follows:

The Max-Min (Max for player 1, Min for player 2) is determined both for pure and
for mixed strategies. The two are identical, — i.e. this variant is strictly determined.
The good strategies which are obtained in this way are rather similar to those of our
Figure 46. Accordingly the characteristics of “Bluffing” do not appear as clearly as in
our Figures 40 and 45. Cf. the analogous considerations in the text above.

CHAPTER V

ZERO-SUM THREE-PERSON GAMES

20. Preliminary Survey
20.1. General Viewpoints

20.1.1. The theory of the zero-sum two-person game having been
completed, we take the next step in the sense of 12.4.: We shall establish
the theory of the zero-sum three-person game. This will bring entirely
new viewpoints into play. The types of games discussed thus far have
had also their own characteristic problems. We saw that the zero-sum
one-person game was characterized by the emergence of a maximum problem
and the zero-sum two-person game by the clear cut opposition of interest
which could no longer be described as a maximum problem. And just as
the transition from the one-person to the zero-sum two-person game
removed the pure maximum character of the problem, so the passage from
the zero-sum two-person game to the zero-sum three-person game obliterates
the pure opposition of interest.

20.1.2. Indeed, it is apparent that the relationships between two players
in a zero-sum three-person game can be manifold. In a zero-sum two-
person game anything one player wins is necessarily lost by the other and
vice versa , — so there is always an absolute antagonism of interests. In a
zero-sum three-person game a particular move of a player — which, for the
sake of simplicity, we assume to be clearly advantageous to him — may be
disadvantageous to both other players, but it may also be advantageous
to one and {a fortiori) disadvantageous to the other opponent. 1 Thus some
players may occasionally experience a parallelism of interests and it may be
imagined that a more elaborate theory will have to decide even whether
this parallelism is total, or partial, etc. On the other hand, opposition of
interest must also exist in the game (it is zero-sum) — and so the theory
will have to disentangle the complicated situations which may ensue.

It may happen, in particular, that a player has a choice among various
policies: That he can adjust his conduct so as to get into parallelism of
interest with another player, or the opposite; that he can choose with
which of the other two players he wishes to establish such a parallelism, and
(possibly) to what extent.

1 All this, of course, is subject to all the complications and difficulties which we have
already recognized and overcome in the zero-sum two-person game: whether a particular
move is advantageous or disadvantageous to a certain player may not depend on that
move alone, but also on what other players do. However, we are trying first to isolate
the new difficulties and to analyze them in their purest form. Afterward we shall dis-
cuss the interrelation with the old difficulties.

220

PRELIMINARY SURVEY

221

20.1.3. As soon as there is a possibility of choosing with whom to
establish parallel interests, this becomes a case of choosing an ally. When
alliances are formed, it is to be expected that some kind of a mutual under-
standing between the two players involved will be necessary. One can
also state it this way: A parallelism of interests makes a cooperation
desirable, and therefore will probably lead to an agreement between the
players involved. An opposition of interests, on the other hand, requires
presumably no more than that a player who has elected this alternative
act independently in his own interest.

Of all this there can be no vestige in the zero-sum two-person game.
Between two players, where neither can win except (precisely) the other's
loss, agreements or understandings are pointless. 1 This should be clear by
common sense. If a formal corroboration (proof) be needed, one can
find it in our ability to complete the theory of the zero-sum two-person
game without ever mentioning agreements or understandings between
players.

20.2. Coalitions

20.2.1. We have thus recognized a qualitatively different feature of the
zero-sum three-person game (as against the zero-sum two-person game).
Whether it is the only one is a question which can be decided only later.
If we succeed in completing the theory of the zero-sura three-person game
without bringing in any further new concepts, then we can claim to have
established this uniqueness. This will be the case essentially when we reach
23.1. For the moment we simply observe that this is a new major element
in the situation, and we propose to discuss it fully before taking up anything
else.

Thus we wish to concentrate on the alternatives for acting in cooperation
with, or in opposition to, others, among which a player can choose. I.e. we
want to analyze the possibility of coalitions — the question between which
players, and against which player, coalitions will form. 2

1 This is, of course, different in a general two-person game (i.e. one with variable sum):
there the two players may conceivably cooperate to produce a greater gain. Thus there
is a certain similarity between the general two-person game and the zero-sum three-
person game.

We shall see in Chap. XI, particularly in 56.2.2., that there is a general connection
behind this: the general n-person game is closely related to the zero-sum n -f 1 -person
game.

* The following seems worth noting : coalitions occur first in a zero-sum game when
the number of participants in the game reaches three. In a two-person game there are
not enough players to go around: a coalition absorbs at least two players, and then
nobody is left to oppose. But while the three-person game of itself implies coalitions,
the scarcity of players is still such as to circumscribe these coalitions in a definite way : a
coalition must consist of precisely two players and be directed against precisely one (the
remaining) player.

If there are four or more players, then the situation becomes considerably more
involved, — several coalitions may form, and these may merge or oppose each other, etc.
Some instances of this appear at the end of 36.1.2., et seq., the end of 37.1.2., et seq.;
another allied phenomenon at the end of 38.3.2.

222

ZERO-SUM THREE-PERSON GAMES

Consequently it is desirable to form an example of a zero-sum three-
person game in which this aspect is foremost and all others are suppressed;
i.e., a game in which the coalitions are the only thing that matters, and the
only conceivable aim of all players . 1

20.2.2. At this point we may mention also the following circumstance:
A player can at best choose between two possible coalitions, since there are
two other players either of whom he may try to induce to cooperate with him
against the third. We shall have to elucidate by the study of the zero-sum
three-person game just how this choice operates, and whether any particular
player has such a choice at all. If, however, a player has only one possi-
bility of forming a coalition (in whatever way we shall in fine interpret this
operation) then it is not quite clear in what sense there is a coalition at all:
moves forced upon a player in a unique way by the necessities of the rules
of the game are more in the nature of a (one sided) strategy than of a (cooper-
ative) coalition. Of course these considerations are rather vague and
uncertain at the present stage of our analysis. We bring them up neverthe-
less, because these distinctions will turn out to be decisive.

It may also seem uncertain, at this stage at least, how the possible
choices of coalitions which confront one player are related to those open to
another; indeed, whether the existence of several alternatives for one player
implies the same for another.

21. The Simple Majority Game of Three Persons
21.1. Description of the Game

21.1. We now formulate the example mentioned above: a simple zero-
sum three-person game in which the possibilities of understandings — i.e.
coalitions — between the players are the only considerations which matter.

This is the game in question :

Each player, by a personal move, chooses the number of one of the
two other players . 2 3 Each one makes his choice uninformed about the
choices of the two other players.

After this the payments will be made as follows: If two players have
chosen each others numbers we say that they form a couple* Clearly

1 This is methodically the same device as our consideration of Matching Pennies in
the theory of the zero-sum two-person game. We had recognized in 14.7.1. that the
decisive new feature of the zero-sum two-person game was the difficulty of deciding
which player “finds out” his opponent. Matching Pennies was the game in which this
“finding out” dominated the picture completely, where this mattered and nothing else.

* Player 1 chooses 2 or 3, player 2 chooses 1 or 3, player 3 chooses 1 or 2.

3 It will be seen that the formation of a couple is in the interest of the players who
create it. Accordingly our discussion of understandings and coalitions in the paragraphs
which follow will show that the players combine into a coalition in order to be able to
form a couple. The difference between the concepts of a couple and a coalition neverthe-
less should not be overlooked: A couple is a formal concept which figures in the set of rules
of the game which we define now; a coalition is a notion belonging to the theory concern-
ing this game (and, as will be seen, many other games).

THE SIMPLE MAJORITY GAME

223

there will be precisely one couple, or none at all. 1 * 2 If there is precisely one
couple, then the two players who belong to it get one-half unit each, while
the third (excluded) player correspondingly loses one unit. If there is no
couple, then no one gets anything. 3

The reader will have no difficulty in recognizing the actual social proc-
esses for which this game is a highly schematized model. We shall call it
the simple majority game (of three players).

21.2. Analysis of the Game. Necessity of “ Understandings”

21.2.1. Let us try to understand the situation which exists when the
game is played.

To begin with, it is clear that there is absolutely nothing for a player
to do in this game but to look for a partner, — i.e. for another player who
is prepared to form a couple with him. The game is so simple and abso-
lutely devoid of any other strategic possibilities that there just is no occasion
for any other reasoned procedure. Since each player makes his personal
move in ignorance of those of the others, no collaboration of the players
can be established during the course of the play. Two players who wish to
collaborate must get together on this subject before the play, — i.e. outside
the game. The player who (in making his personal move) lives up to his
agreement (by choosing the partner's number) must possess the conviction
that the partner too will do likewise. As long as we are concerned only
with the rules of the game, as stated above, we are in no position to judge
what the basis for such a conviction may be. In other words what, if
anything, enforces the “ sanctity" of such agreements? There may be
games which themselves — by virtue of the rules of the game as defined in
6.1. and 10.1. — provide the mechanism for agreements and for their enforce-
ment. 4 But we cannot base our considerations on this possibility, since a
game need not provide this mechanism; the simple majority game described
above certainly does not. Thus there seems to be no escape from the
necessity of considering agreements concluded outside the game. If we do
not allow for them, then it is hard to see what, if anything, will govern the
conduct of a player in a simple majority game. Or, to put this in a some-
what different form:

1 I.e. there cannot be simultaneously two different couples. Indeed, two couples
must have one player on common (since there are only three players), and the number
chosen by this player must be that of the other player in both couples, — i.e. the two
couples are identical.

* It may happen that no couples exist: e.g., if 1 chooses 2, 2 chooses 3, and 3 chooses 1.

3 For the sake of absolute formal correctness this should still be arranged according
to the patterns of 6. and 7. in Chap. II. We leave this to the reader, as in the analogous
situation discussed in footnote 1 on p. 191.

4 By providing personal moves of one player, about which only one other player is
informed and which contain (possibly conditional) statements of the first player's future
policy; and by prescribing for him to adhere subsequently to these statements, or by
providing (in the functions which determine the outcome of a game) penalties for the
non-adherence.

224

ZERO-SUM THREE-PERSON GAMES

We are trying to establish a theory of the rational conduct of the partici-
pants in £ given game. In our consideration of the simple majority game
we have reached the point beyond which it is difficult to go in formulating
such a theory without auxiliary concepts such as “agreements,” “under-
standings,” etc. On a later occasion we propose to investigate what theo-
retical structures are required in order to eliminate these concepts. For
this purpose the entire theory of this book will be required as a foundation,
and the investigation will proceed along the lines indicated in Chapter XII,
and particularly in 66. At any rate, at present our position is too weak and
our theory not sufficiently advanced to permit this “self-denial.” We shall
therefore, in the discussions which follow, make use of the possibility of the
establishment of coalitions outside the game; this will include the hypothesis
that they are respected by the contracting parties.

21 . 2 . 2 . These agreements have a certain amount of similarity with
“conventions” in some games like Bridge — with the fundamental difference,
however, that those affected only one “organization” (i.e. one player split
into two “persons”) while we are now confronted with the relationship of
two players. At this point the reader may reread with advantage our
discussion of “conventions” and related topics in the last part of 6.4.2. and
6.4.3., especially footnote 2 on p. 53.

21 . 2 . 3 . If our theory were applied as a statistical analysis of a long series
of plays of the same game — and not as the analysis of one isolated play — an
alternative interpretation would suggest itself. We should then view
agreements and all forms of cooperation as establishing themselves by
repetition in such a long series of plays.

It would not be impossible to derive a mechanism of enforcement from
the player's desire to maintain his record and to be able to rely on the record
of his partner. However, we prefer to view our theory as applying to an
individual play. But these considerations, nevertheless, possess a certain
significance in a virtual sense. The situation is similar to the one which
we encountered in the analysis of the (mixed) strategies of a zero-sum two-
person game. The reader should apply the discussions of 17.3. mutatis
mutandis to the present situation.

21.3. Analysis of the Game : Coalitions. The Role of Symmetry

21 . 3 . Once it is conceded that agreements may exist between the players
in the simple majority game, the path is clear. This game offers to players
who collaborate an infallible opportunity to win — and the game does not
offer to anybody opportunities for rational action of any other kind. The
rules are so elementary that this point ought to be fully convincing.

Again the game is wholly symmetric with respect to the three players.
That is true as far as the rules of the game are concerned : they do not offer
to any player any possibility which is not equally open to any other player.
What the players do within these possibilities is, of course, another matter.
Their conduct may be unsymmetric; indeed, since understandings, i.e. coali-

FURTHER EXAMPLES

225

tions, are sure to arise, it will of necessity be unsymmetric. Among
the three players there is room for only one coalition (of two players) and
one player will necessarily be left out. It is quite instructive to observe how
the rules of the game are absolutely fair (in this case, symmetric), but the
conduct of the players will necessarily not be. 1 * 2 *

Thus the only significant strategic feature of this game is the possibility
of coalitions between two players. 8 And since the rules of the game are
perfectly symmetrical, all three possible coalitions 4 must be considered
on the same footing. If a coalition is formed, then the rules of the game
provide that the two allies get one unit from the third (excluded) player —
each one getting one-half unit.

Which of these three possible coalitions will form, is beyond the scope
of the theory, — at least at the present stage of its development. (Cf . the end
of 4.3.2.) We can say only that it would be irrational if no coalitions were
formed at all, but as to which particular coalition will be formed must depend
on conditions which we have not yet attempted to analyze.

22. Further Examples

22.1. Unsymmetric Distribution. Necessity of Compensations

22.1.1. The remarks of the preceding paragraphs exhaust, at least for
the time being, the subject of the simple majority game. We must now
begin to remove, one by one, the extremely specializing assumptions which
characterized this game: its very special nature was essential for us in
order to observe the role of coalitions in a pure and isolated form — in vitro —

1 We saw in 17.11.2. that no such thing occurs in the zero-sum two-person games.
There, if the rules of the game are symmetric, both players get the same amount (i.e.
the value of the game is zero), and both have the same good strategies. I.e. there is no
reason to expect a difference in their conduct or in the results which they ultimately
obtain.

It is on emergence of coalitions — when more than two players are present — and of
the “squeeze" which they produce among the players, that the peculiar situation
described above arises. (In our present case of three players the “squeeze" is due to the
fact that each coalition can consist of only two players, i.e. less than the total number of
players but more than one-half of it. It would be erroneous, however, to assume that no
such “squeeze" obtains for a greater number of players.)

* This is, of course, a very essential feature of the most familiar forms of social organi-
zations. It is also an argument which occurs again and again in the criticism directed
against these institutions, most of all against the hypothetical order based upon “fowser
faire ." It is the argument that even an absolute, formal fairness — symmetry of the rules
of the game — does not guarantee that the use of these rules by the participants will be
fair and symmetrical. Indeed, this “does not guarantee" is an understatement: it is to
be expected that any exhaustive theory of rational behavior will show that the partici-
pants are driven to form coalitions in unsymmetric arrangements.

To the extent to which an exact theory of these coalitions is developed, a real under-
standing of this classical criticism is achieved. It seems worth emphasizing that this
characteristically “social" phenomenon occurs only in the case of three or more
participants.

5 Such a coalition is in this game, of course, simply an agreement to choose each other's
numbers, so as to form a couple in the sense of the rules. This situation was foreseen
already at the beginning of 4.3.2.

4 Between players 1,2; 1,3; 2,3.

226

ZERO-SUM THREE-PERSON GAMES

but now this step is completed. We must begin to adjust our ideas to
more general situations.

22.1.2. The specialization which we propose to remove first is this: In
the simple majority game any coalition can get one unit from the opponent;
the rules of the game provide that this unit must be divided evenly
among the partners. Let us now consider a game in which each coalition
offers the same total return, but where the rules of the game provide for a
different distribution. For the sake of simplicity let this be the case only in
the coalition of players 1 and 2, where player 1, say, is favored by an amount e.
The rules of the modified game are therefore as follows:

The moves are the same as in the simple majority game described in
21.1. The definition of a couple is the same too. If the couple 1,2 forms,
then player 1 gets the amount £ + e l , player 2 gets the amount £ — e, and
player 3 loses one unit. If any other couple forms (i.e. 1,3 or 2,3) then
the two players which belong to it get one-half unit each while the third
(excluded) player loses one unit.

What will happen in this game?

To begin with, it is still characterized by the possibility of three coalitions
— corresponding to the three possible couples — which may arise in it.
Prima facie it may seem that player 1 has an advantage, since at least in his
couple with player 2 he gets more by e than in the original, simple majority
game.

However, this advantage is quite illusory. If player 1 would really
insist on getting the extra c in the couple with player 2, then this would
have the following consequence: The couple 1,3 would never form, because
the couple 1,2 is more desirable from Ts point of view; the couple 1,2 would
never form, because the couple 2,3 is more desirable from 2’s point of view;
but the couple 2,3 is entirely unobstructed, since it can be brought about
by a coalition of 2,3 who then need pay no attention to 1 and his special
desires. Thus the couple 2,3 and no other will form; and player 1 will not
get £ + € nor even one-half unit, but he will certainly be the excluded player
and lose one unit.

So any attempt of player 1 to keep his privileged position in the couple
1,2 is bound to lead to disaster for him. The best he can do is to take
steps which make the couple 1,2 just as attractive for 2 as the competing
couple 2,3. That is to say, he acts wisely if, in case of the formation of
a couple with 2, he returns the extra t to his partner. It should be noted
that he cannot keep any fraction of e; i.e., if he should try to keep an extra
amount e' for himself, 2 then the above arguments could be repeated literally
with «' in place of e. 3

1 It seems natural to assume 0 < « <

* We mean of course 0 <«'<€.

* So the motives for player l’s ultimate disaster — the certain formation of couple
2,3 — would be weaker, but the disaster the same and just as certain as before. Of. in
this connection footnote 1 on p. 228.

FURTHER EXAMPLES

227

22.1.3. One could try some other variations of the original, simple,
majority game, still always maintaining that the total value of each coalition
is one unit. E.g. we could consider rules where player 1 gets the amount
i + € in each couple 1,2, 1,3; while players 2 and 3 split even in the couple
2,3. In this case neither 2 nor 3 would care to cooperate with 1 if 1 should
try to keep his extra e or any fraction thereof. Hence any such attempt
of player 1 would again lead with certainty to a coalition of 2,3 against him
and to a loss of one unit.

Another possibility would be that two players are favored in all couples
with the third: e.g. in the couples 1,3 and 2,3, players 1 and 2 respectively
get i + « while 3 gets only i — e; and in the couple 1,2 both get one-half
unit each. In this case both players 1 and 2 would lose interest in a coalition
with each other, and player 3 will become the desirable partner for each of
them. One must expect that this will lead to a competitive bidding for his
cooperation. This must ultimately lead to a refund to player 3 of the extra
advantage e. Only this will bring the couple 1,2 back into the field of
competition and thereby restore equilibrium.

22.1.4. We leave to the reader the consideration of further variants,
where all three players fare differently in all three couples. Furthermore
we shall not push the above analysis further, although this could be done
and would even be desirable in order to answer some plausible objections.
We are satisfied with having established some kind of a general plausibility
for our present approach which can be summarized as follows: It seems
that what a player can get in a definite coalition depends not only on what
the rules of the game provide for that eventuality, but also on the other
(competing) possibilities of coalitions for himself and for his partner. Since
the rules of the game are absolute and inviolable, this means that under
certain conditions compensations must be paid among coalition partners;
i.e. that a player must have to pay a well-defined price to a prospective
coalition partner. The amount of the compensations will depend on what
other alternatives are open to each of the players.

Our examples above have served as a first illustration for these principles.
This being understood, we shall now take up the subject de novo and in
more generality, and handle it in a more precise manner. 1

22.2. Coalitions of Different Strength. Discussion

22.2.1. In accordance with the above we now take a far reaching step
towards generality. We consider a game in which this is the case:

If players 1,2 cooperate, then they can get the amount c, and no more,
from player 3; if players 1,3 cooperate, they can get the amount 6, and no
more, from player 2; if players 2,3 cooperate, they can get the amount a, and
no more, from player 1.

1 This is why we need not analyze any further the heuristic arguments of this para-
graph — the discussion of the next paragraphs takes care of everything.

All these possibilities were anticipated at the beginning of 4.3.2. and in 4.3.3.

228

ZERO-SUM THREE-PERSON GAMES

We make no assumptions whatsoever concerning further particulars
about the rules of this game. So we need not describe by what steps — of
what order of complication — the above amounts are secured. Nor do we
state how these amounts are divided between the partners, whether and how
either partner can influence or modify this distribution, etc.

We shall nevertheless be able to discuss this game completely. But
it will be necessary to remember that a coalition is probably connected
with compensations passing between the partners. The argument is as
follows:

22 . 2 . 2 . Consider the situation of player 1 . He can enter two alternative
coalitions: with player 2 or with player 3. Assume that he attempts to
retain an amount x under all conditions. In this case player 2 cannot
count upon obtaining more than the amount c — x in a coalition with
player 1 . Similarly player 3 cannot count on getting more than the
amount 6 — x in a coalition with player 1. Now if the sum of these upper
bounds — i.e. the amount (c — x) + (b — x) — is less than what players 2
and 3 can get by combining with each other in a coalition, then we may
safely assume that player 1 will find no partner. 1 A coalition of 2 and 3
can obtain the amount a. So we see: If player 1 desires to get an amount x
under all conditions, then he is disqualified from any possibility of finding
a partner if his x fulfills

(c — x) + (b — x) < a.

I.e. the desire to get x is unrealistic and absurd unless
(c — x) + (6 — x) ^ a.

This inequality may be written equivalently as

— a + b + c

x < —

2

We restate this:

(22:1 :a) Player 1 cannot reasonably maintain a claim to get under

all conditions more than the amount a = — Q c .

The same considerations may be repeated for players 2 and 3, and they
give:

(22:1 :b) Player 2 cannot reasonably maintain a claim to get under

all conditions more than the amount 0 = •

(22:1 :c) Player 3 cannot reasonably maintain a claim to get under

a + b — c

all conditions more than the amount y =

2

1 We assume, of course, that a player is not indifferent to any possible profit, however
small. This was implicit in our discussion of the zero-sum two-person game as well.

The traditional idea of the u homo oeconomicu to the extent to which it is clearly
conceived at all, also contains this assumption.

FURTHER EXAMPLES

229

22 . 2 . 3 . Now the criteria (22:l:a)-(22:l:c) were only necessary ones, and
one could imagine a priori that further considerations could further lower
their upper bounds, a, 0, 7 — or lead to some other restrictions of what
the players can aim for. This is not so, as the following simple con-
sideration shows.

One verifies immediately that

a + 0 = c, a + 7 = &, 0 + 7= a.

In other words: If the players 1 , 2,3 do not aim at more than permitted by
(22:1 :a), (22:1 :b), (22:1 :c), i.e. than a, 0, 7 respectively, then any two
players who combine can actually obtain these amounts in a coalition.
Thus these claims are fully justified. Of course only two players — the two
who form a coalition — can actually obtain their “justified” dues. The
third player, who is excluded from the coalition, will not get a, 0, 7 respec-
tively, but —a, — 6, — c instead. 1

22.3. An Inequality. Formulae

22 . 3 . 1 . At this point an obvious question presents itself: Any player

1 . 2.3 can get the amount a, 0, 7 respectively if he succeeds in entering a
coalition; if he does not succeed, he gets instead only —a, — 6, — c. This
makes sense only if a, 0, 7 are greater than the corresponding —a, — 6, — c,
since otherwise a player might not want to enter a coalition at all, but
might find it more advantageous to play for himself. So the question is
whether the three differences

p = a — ( — a) = a + a,

<7 = 0- (~b) =0 + 6,
r = 7 ~ (-c) = y + c,

are all ^ 0.

It is immediately seen that they are all equal to each other. Indeed :

CL + 6 + c

p = q = r — £

We denote this quantity by A/ 2 . Then our question is whether

A=a + 6 + c^0.

This inequality can be demonstrated as follows:

22 . 3 . 2 . A coalition of the players 1,2 can obtain (from player 3 ) the
amount c and no more. If player 1 plays alone, then he can prevent players

2.3 from reducing him to a result worse than —a since even a coalition of
players 2,3 can obtain (from player 1) the amount +a and no more; i.e.
player 1 can get the amount —a for himself without any outside help.
Similarly, player 2 can get the amount —6 for himself without any outside
help. Consequently the two players 1,2 between them can get the amount

1 These are indeed the amounts which a coalition of the other players can wrest from
players 1,2,3 respectively. The coalition cannot take more.

230

ZERO-SUM THREE-PERSON GAMES

— (a -f 6 ) even if they fail to cooperate with each other. Since the maxi-
mum they can obtain together under any conditions is c, this implies
c ^ —a — b i.e. A = o + b + c^0.

22 . 3 . 3 . This proof suggests the following remarks:

First: We have based our argument on player 1 . Owing to the sym-
metry of the result A=a+ 6 +c ^0 with respect to the three players,
the same inequality would have obtained if we had analyzed the situation
of player 2 or player 3. This indicates that there exists a certain symmetry
in the role of the three players.

Second: A = 0 means c = —a — b or just as well a = — a, and the two
corresponding pairs of equations which obtain by the cyclic permutation
of the three players. So in this case no coalition has a raison d'&tre: Any
two players can obtain, without cooperating, the same amount which they
can produce in perfect cooperation (e.g. for players 1 and 2 this amount is
'—a — b = c). Also, after all is said and done, each player who succeeds
in joining a coalition gets no more than he could get for himself without
outside help (e.g. for player 1 this amount is a = —a).

If, on the other hand, A > 0 then every player has a definite interest in
joining a coalition. The advantage contained in this is the same for all
three players : A/2.

Here we have again an indication of the symmetry of certain aspects
of the situation for all players: A/2 is the inducement to seek a coalition; it
is the same for all players.

22 . 3 . 4 . Our result can be expressed by the following table:

Player

1

2

3

Value of a play

With coalition

a

0

7

Without coalition

—a

-b

—c

Figure 49.

If we put

, . 1 A 1 A -2 a + b + c

a = - a + -A = «-gA = 3 >

L , 1 A 0 1 A a - 2b + c

6 "- & + 3 A = 0~6 A== 3 ’

, , 1 A 1 A a + b - 2 c

c = _ c + _ A==7 __ A = 3 ,

then we have

a' + V + c' = 0,

and we can express the above table equivalently in the following manner:

(22 :A) A play has for the players 1,2,3 the basic values a', 6 ', c' respec-

tively. (This is a possible valuation, since the sum of these

THE GENERAL CASE

231

values is zero, cf. above). The play will, however, certainly be
attended by the formation of a coalition. Those two players
who form it get (beyond their basic values) a premium of A/6 and
the excluded player sustains a loss of — A/3.

Thus the inducement to form a coalition is A/2 for each
player, and always A/2 ^ 0.

23. The General Case

23.1. Exhaustive Discussion. Inessential and Essential Games

23.1.1. We can now remove all restrictions.

Let T be a perfectly arbitrary zero-sum three-person game. A simple
consideration suffices to bring it within the reach of the analysis of 22.2.,
22.3. We argue as follows:

If two players, say 1 and 2, decide to cooperate completely — postponing
temporarily, for a later settlement, the question of distribution, i.e. of the
compensations to be paid between partners — then T becomes a zero-sum
two-person game. The two players in this new game are: the coalition 1,2
(which is now a composite player consisting of two “natural persons”),
and the player 3. Viewed in this manner V falls under the theory of the
zero-sum two- person game of Chapter III. Each play of this game has a
well defined value (we mean the v' defined in 17.4.2.). Let us denote by c
the value of a play for the coalition 1,2 (which in our present interpretation
is one of the players).

Similarly we can assume an absolute coalition between players 1,3 and
view T as a zero-sum two-person game between this coalition and the player
2. We then denote by b the value of a play for the coalition 1,3.

Finally we can assume an absolute coalition between players 2,3, and
view T as a zero-sum two-person game between this coalition and the
player 1. We then denote by a the value of a play for the coalition 2,3.

It ought to be understood that we do not — yet ! — assume that any such
coalition will necessarily arise. The quantities a, 6, c are merely computa-
tionally defined; we have formed them on the basis of the main (mathe-
matical) theorem of 17.6. (For explicit expressions of a, 6, c cf. below.)

23.1.2. Now it is clear that the zero-sum three-person game T falls
entirely within the domain of validity of 22.2., 22.3.: a coalition of the
players 1,2 or 1,3 or 2,3 can obtain (from the excluded players 3 or 2 or 1)
the amounts c, 6, a respectively, and no more. Consequently all results of
22.2., 22.3. hold, in particular the one formulated at the end which describes
every player’s situation with and without a coalition.

23.1.3. These results show that the zero-sum three-person game falls
into two quantitatively different categories, corresponding to the possi-
bilities A = 0 and A > 0. Indeed:

A = 0: We have seen that in this case coalitions have no raison d'&re,
and each player can get the same amount for himself, by playing a “lone
hand” against all others, as he could obtain by any coalition. In this case,

232

ZERO-SUM THREE-PERSON GAMES

and in this case alone, it is possible to assume a unique value of each play
for each player, — the sum of these values being zero. These are the basic
values a', c' mentioned at the end of 22.3. In this case the formulae of

22.3. show that a! = a = — a, V = 0 = — b f c' = y = — c. We shall
call a game in this case, in which it is inessential to consider coalitions, an
inessential game.

A > 0: In this case there is a definite inducement to form coalitions, as
discussed at the end of 22.3. There is no need to repeat the description
given there; we only mention that now a > a' > -a, 0 > V > — b,
7 > c' > — c. We shall call a game in this case, in which coalitions are
essential, an essential game.

Our above classification, inessential and essential, applies at present
only to zero-sum three-person games. But we shall see subsequently that it
can be extended to all games and that it is a differentiation of central
importance.

23.2. Complete Formulae

23.2. Before we analyze this result any further, let us make a few purely
mathematical remarks about the quantities a, b, c — and the a, ft, y, a', 6', c',
A based upon them — in terms of which our solution was expressed.

Assume the zero-sum three-person game T in the normalized form of

11.2.3. There the players 1,2,3 choose the variables r h t 2 , t 3 respectively
(each one uninformed about the two other choices) and get the amounts
3 Ci(ti, t 2 , r 3 ), 3C 2 (t!, t 2 , t 3 ), 3C 8 (ti, t 2 , t 3 ) respectively. Of course (the game is
zero-sum) :

3Ci(ti, t 2 , t 3 ) + 3C 2 (n, t 2 , t 3 ) + 3C 3 (n, r 2 , r 3 ) s 0.

The domains of the variables are:

ti = 1, 2, • • • , 01,

t 2 = 1, 2, • • • , 02,

r 3 = 1, 2, • • • , 0 3 .

Now in the two-person game which arises between an absolute coalition of
players 1,2, and the player 3, we have the following situation:

The composite player 1,2 has the variables n, r 2 ; the other player 3
has the variable r 3 . The former gets the amount

3Ci(ti, t 2 , t 3 ) +3C 2 (ti, T 2 , t 3 ) = — 0C 3 (ri, t 2 , t 3 ),
the latter the negative of this amount.

A mixed strategy of the composite player 1,2 is a vector £ of 5*^, the

components of which we may denote by Thus the £ of are

characterized by

* V r, £ 0, X “ 1-

TlSt

1 The number of pairs n, r t is, of course,

DISCUSSION OF AN OBJECTION

233

A mixed strategy of the player 3 is a vector tj of Sp t the components of
which we denote by 77 v The rj of Sfi t are characterized by

Y )r % ^ 0 , Vr t = 1 .

The bilinear form K( £ , 77 ) of (17 :2) in 17.4.1. is therefore

K( £ , rj ) = ^ {3Ci(ri, r 2 , t 3 ) + JC 2 (n, r 2 , r 3 ) ) £ Ti T ^ Tj

'1 r t .r 9

s “ 5) 3C 3 (n, r 2 , r 3 )fr iiTi iy T| ,

r i r 2. T i

and finally

c = Max— Min— K( $ , q ) = Min— Max— K( £ , q ).

£ »7 7 €

The expressions for 6, a obtain from this by cyclical permutations of the
players 1,2,3 in all details of this representation.

We repeat the formulae expressing a, 0, y, a', b\ c' and A:

A = a + b + c necessarily ^ 0,

-a + b + c

a =

— 2 a *4" b -f- c

0 =

a — b + c

V =

a — 2b + c

7 =

<2 -}- 6

a + 6 — 2c

and we have

a = a' +

-a = a -3,

A 2: 0,

a' + 6' + c' = 0,

-6 = 6 '

A

3'

7 = C' +

-c = c -3.

24. Discussion of an Objection

24.1. The Case of Perfect Information and Its Significance

24.1.1. We have obtained a solution for the zero-sum three-person game
which accounts for all possibilities and which indicates the direction that the
search for the solutions of the n- person game must take: the analysis of all
possible coalitions, and the competitive relationship which they bear to
each other, — which should determine the compensations that players who
want to form a coalition will pay to each other.

We have noticed already that this will be a much more difficult problem
for n ^ 4 players than it was for n = 3 (cf. footnote 2, p. 221).

234

ZERO-SUM THREE-PERSON GAMES

Before we attack this question, it is well to pause for a moment to
reconsider our position. In the discussions which follow we shall put the
main stress on the formation of coalitions and the compensations between
the participants in those coalitions, using the theory of the zero-sum two-
person game to determine the values of the ultimate coalitions which oppose
each other after all players have “ taken sides” (cf. 25.1.1., 25.2.). But is
this aspect of the matter really as universal as we propose to claim?

We have adduced already some strong positive arguments for it, in our
discussion of the zero-sum three-person game. Our ability to build the
theory of the n-person game (for all n) on this foundation will, in fine , be
the decisive positive argument. But there is a negative argument — an
objection — to be considered, which arises in connection with those games
where perfect information prevails.

The objection which we shall now discuss applies only to the above
mentioned special category of games. Thus it would not, if found valid,
provide us with an alternative theory that applies to all games. But since
we claim a general validity for our proposed stand, we must invalidate all
objections, even those which apply only to some special case. 1

24.1.2. Games with perfect information have already been discussed in
15. We saw there that they have important peculiarities and that their
nature can be understood fully only when they are considered in the extensive
form — and not merely in the normalized one on which our discussion chiefly
relied (cf. also 14.8.).

The analysis of 15. began by considering n- person games (for all n),
but in its later parts we had to narrow it to the zero-sum two-person game.
At the end, in particular, we found a verbal method of discussing it (cf. 15.8.)
which had some remarkable features: First, while not entirely free from
objections, it seemed worth considering. Second, the argumentation used
was rather different from that by which we had resolved the general case of
the zero-sum two-person game — and while applicable only to this special
case, it was more straight forward than the other argumentation. Third,
it led — for the zero-sum two-person games with perfect information — to
the same result as our general theory.

Now one might be tempted to use this argumentation for n ^ 3 players
too; indeed a superficial inspection of the pertinent paragraph 15.8.2. does
not immediately disclose any reason why it should be restricted (as there) to
n = 2 players (cf., however, 15.8.3.). But this procedure makes no men-
tion of coalitions or understandings between players, etc.; so if it is usable
for n = 3 players, then our present approach is open to grave doubts. 2 We

1 In other words: in claiming general validity for a theory one necessarily assumes the
burden of proof against all objectors.

* One might hope to evade this issue, by expecting to find A = 0 for all zero-sum three-
person games with perfect information. This would make coalitions unnecessary. Cf.
the end of 23.1.

Just as games with perfect information avoided the difficulties of the theory of zero-
sum two-person games by being strictly determined (cf. 15.6.1.), they would now avoid
those of the zero-sum three-person games by being inessential.

DISCUSSION OF AN OBJECTION

235

propose to show therefore why the procedure of 15.8. is inconclusive when
the number of players is three or more.

To do this, let us repeat some characteristic steps of the argumentation
in question (cf. 15.8.2., the notations of which we are also using).

24.2. Detailed Discussion. Necessity of Compensations between Three or More Players

24.2.1. Consider accordingly a game T in which perfect information
prevails. Let 3Tli, 3112, * * • , 3Tl„ be its moves, a i, 02, • • * , <r, the choices
connected with these moves, x(<n, • • • , a,) the play characterized by
these choices, and h • • • , <r„)) the outcome of this play for the player
j(= 1, 2, • • • , n).

Assume that the moves 3Tli, 3TC 2 , • • * , 3Tl„_i have already been made,
the outcome of their choices being <r 1, <7 2 , • ■ * , <r„_i and consider the last
move 3TC„ and its <r P . If this is a chance move — i.e. k,(<r h • • • , <r,_i) = 0, —
then the various possible values a, = 1,2, • • • , a„(<r 1, • • • , <r„_i) have the
probabilities p„(l), p y (2), • • • , p„(a„(<7 1, • • ■ , cr„_i)), respectively. If this
is a personal move of player A: — i.e. A: v (cri, • • • , <r„_ 1) = k = 1,2, • • • , n, —
then player k will choose <7, so as to make CF* (tt(cti, • • • , <7,-. x , <r,)) a maxi-
mum. Denote this <r, by <r,(<ri, • • • , <7,-1). Thus one can argue that the
value of the play is already known (for each player j = 1, • • • n) after
the moves 3Tlj, 3Tl 2 , • • • , (and before 3TI,!), — i.e. as a function of

(7 1, a 2 , • • • , <7„_i alone. Indeed: by the above it is

1 a p (c lt • * • , 9 *-|)

( = 2) ’ ’ ’ , <r-i, <r,))

i» ' - ’ , S , '" 1

/ for fc r (ffi, • • • , = 0,

' = Sj(ir(<n, • • • , <J,{a i, • • • , <r,_i))),

where <7, = <7^(<7i, ••• , <7,_i) maximizes
$k(ir(<T 1, • • • , <7,- 1, <r,)) for
k P {<Ti, f o P — 1 ) =s A = 1, • • • ,71.

Consequently we can treat the game T as if it consisted of the moves
3Tli, 3R 2 , • • • , 3Tl,_i only (without 3TC,).

By this device we have removed the last move 3Tl„. Repeating it, we
can similarly remove successively the moves 3TC,_i, 3Tl,_ 2 , • • • , 3Tl 2 , 3Hi and
finally obtain a definite value of the play (for each player j = 1, 2, • • • n).

24.2.2. For a critical appraisal of this procedure consider the last two
steps 3H,_i, 3TI, and assume that they are personal moves of two different

This, however, is not the case. To see that, it suffices to modify the rules of the
simple majority game (cf. 21.1.) as follows: Let the players 1,2,3 make their personal
moves (i.e. the choices of n, rj, r% respectively, cf. loc. cit.) in this order, each one being
informed about the anterior moves. It is easy to verify that the values c, 6, a of th o
three coalitions 1,2, 1,3, 2,3 are the same as before

c » 6 — o — 1, A*o-f5+c*3>0.

A detailed discussion of this game, with particular respect to the considerations of
21.2., would be of a certain interest, but we do not propose to continue this subject
further at present.

236

ZERO-SUM THREE-PERSON GAMES

players, say 1,2 respectively. In this situation we have assumed that
player 2 will certainly choose 0, so as to maximize $2(0-1, * * * , 0,-1, 0,).
This gives a <r> = 0,(01, • • • , o v - 1). Now we have also assumed that
player 1, in choosing 0,-1 can rely on this; i.e. that he may safely replace the
$1(0-1, • • • , 0^-1, 0-,), (which is what he will really obtain), by
$1(0-1, • • • , 0-„_i, 0,(01, * * * , 0,-1)) and maximize this latter quantity. 1
But can he rely on this assumption?

To begin with, 0,(01, • • • , 0,-1) may not even be uniquely determined:
$2(01, • • • , 0,-1, 0,) may assume its maximum (for given 01, • • • , 0,-1)
at several places 0,. In the zero-sum two-person game this was irrelevant:
there $1 = —$2, hence two 0, which give the same value to $ 2 , also give the
same value to $i. 2 But even in the zero-sum three-person game, $2 does not
determine $1, due to the existence of the third player and his $3! So it
happens here for the first time that a difference which is unimportant for
one player may be significant for another player. This was impossible in
the zero-sum two-person game, where each player won (precisely) what the
other lost.

What then must player 1 expect if two 0, are of the same importance for
player 2, but not for player 1 ? One must expect that he will try to induce
player 2 to choose the 0, which is more favorable to him. He could offer
to pay to player 2 any amount up to the difference this makes for him.

This being conceded, one must envisage that player 1 may even try to
induce player 2 to choose a 0, which does not maximize $2(01, * • • , 0,-1, 0,).
As long as this change causes player 2 less of a loss than it causes player 1
a gain, 3 player 1 can compensate player 2 for his loss, and possibly even
give up to him some part of his profit.

24.2.3. But if player 1 can offer this to player 2, then he must also count
on similar offers coming from player 3 to player 2. I.e. there is no certainty
at all that player 2 will, by his choice of 0,, maximize $2(01, • * * , 0,-1, 0,).
In comparing two 0, one must consider whether player 2’s loss is over-
compensated by player Vs or player 3's gain, since this could lead to under-
standings and compensations. I.e. one must analyze whether a coalition 1,2
or 2,3 would gain by any modification of 0,.

24.2.4. This brings the coalitions back into the picture. A closer analysis
would lead us to the considerations and results of 22.2., 22.3., 23. in every
detail. But it does not seem necessary to carry this out here in complete
detail: after all, this is just a special case, and the discussion of 22.2., 22.3.,
23. was of absolutely general validity (for the zero-sum three-person game)

1 Since this is a function of <n, • • • , o-„_ 2 , <r,_ 1 only, of which • • • , are known

at and <r,,_i is controlled by player 1, he is able to maximize it.

He cannot in any sense maximize 5 i(<r i, • • * , <7, ) since that also depends on <r,

which he neither knows nor controls.

2 Indeed, we refrained in 15.8.2. from mentioning $ 2 at all: instead of maximizing $2,
we talked of minimizing $1. There was no need even to introduce 0,(0 lf • * • , 0,-1)
and everything was described by Max and Min operations on SFi.

2 I.e. when it happens at the expense of player 3.

DISCUSSION OF AN OBJECTION

237

provided that the consideration of understandings and compensations, i.e. of
coalitions, is permitted.

We wanted to show that the weakness of the argument of 15.8.2., already
recognized in 15.8.3., becomes destructive exactly when we go beyond the
zero-sum two-person games, and that it leads precisely to the mechanism of
coalitions etc. foreseen in the earlier paragraphs of this chapter. This
should be clear from the above analysis, and so we can return to our original
method in dealing with zero-sum three-person games, — i.e. claim full validity
for the results of 22.2., 22.3., 23.

CHAPTER VI

FORMULATION OF THE GENERAL THEORY:

ZERO-SUM n-PERSON GAMES

25. The Characteristic Function
25.1. Motivation and Definition

25.1.1. We now turn to the zero-sum n- person game for general n. The
experience gained in Chapter V concerning the case n = 3 suggests that
the possibilities of coalitions between players will play a decisive role in the
general theory which we are developing. It is therefore important to
evolve a mathematical tool which expresses these “possibilities” in a
quantitative way.

Since we have an exact concept of “value” (of a play) for the zero-sum
two-person game, we can also attribute a “value” to any given group of
players, provided that it is opposed by the coalition of all the other players.
We shall give these rather heuristic indications an exact meaning in what
follows. The important thing is, at any rate, that we shall thus reach a
mathematical concept on which one can try to base a general theory — and
that the attempt will, in fine y prove successful.

Let us now state the exact mathematical definitions which carry out this
program.

25.1.2. Suppose then that we have a game r of n players who, for the
sake of brevity, will be denoted by 1, 2, • • • , n. It is convenient to
introduce the set / = (1, 2, • • • , n) of all these players. Without yet
making any predictions or assumptions about the course a play of this game
is likely to take, we observe this: if we group the players into two parties,
and treat each party as an absolute coalition — i.e. if we assume full coopera-
tion within each party — then a zero-sum two-person game results. 1 Pre-
cisely: Let S be any given subset of /, — S its complement in /. We
consider the zero-sum two-person game which results when all players k
belonging to S cooperate with each other on the one hand, and all players k
belonging to — S cooperate with each other on the other hand.

Viewed in this manner T falls under the theory of the zero-sum two-
person game of Chapter III. Each play of this game has a well defined
value (we mean the v' defined in 17.8.1.). Let us denote by v(S) the value
of a play for the coalition of all k belonging to S (which, in our present inter-
pretation, is one of the players).

1 This is exactly what we did in the case n ** 3 in 23.1.1. The general possibility was
already alluded to at the beginning of 24.1.

238

THE CHARACTERISTIC FUNCTION

239

Mathematical expressions for v(S) obtain as follows: 1
25.1.3. Assume the zero-sum n- person game T in the normalized form of
11.2.3. There each player k = 1, 2, • • • , n chooses a variable r* (each one
uninformed about the n — 1 other choices) and gets the amount

3Cfc(Tl, T 2 , * * * , T n ).

Of course (the game is zero-sum) :

(25:1) £ 3C*(r„ • ■ • , r.) - 0.

Jfc-1

The domains of the variables are:

r k = 1, • • • , fa for k = 1, 2, • • • , n.

Now in the two-person game which arises between an absolute coalition of all
players k belonging to S (player 1') and that one of all players k belonging
to — S (player 2'), we have the following situation:

The composite player 1' has the aggregate of variables r* where k runs
over all elements of S. It is necessary to treat this aggregate as one variable
and we shall therefore designate it by one symbol t 8 . The composite
player 2' has the aggregate of variables r k where k runs over all elements
of —S. This aggregate too is one variable, which we designate by the
symbol t~ s . The player 1' gets the amount

(25:2) 3C(r s , t~ s ) = £ 3C *(n, • • • , r n ) = — £ 3C*(ri, • • ■ , r n ); 2

k in S kin - S

the player 2' gets the negative of this amount.

A mixed strategy of the player 1' is a vector £ of S fiSy z the components

— y

of which we denote by £ T s. Thus the ( of Sfi are characterized by

(rs £ 0, 2 = 1.

T*

A mixed strategy of the player 2' is a vector rj of Sp-a, 4 the components
of which we denote by ry T -«. Thus the rj of Sp-a are characterized by

Vr s ^0, 5) rjr-a = 1.

r-«

1 This is a repetition of the construction of 23.2., which applied only to the special
case n « 3.

* The t 5 , t~ 8 of the first expression form together the aggregate of the n, • • • , r„ of
the two other expressions; so r 5 , r“ s determine those n, * • • , r n .

The equality of the two last expressions is, of course, only a restatement of the
zero-sum property.

3 p 3 is the number of possible aggregates t s , i.e. the product of all 0* where k runs over
all elements of S .

4 P~ 3 is the number of possible aggregates r“ s , i.e. the product of all 0* where k runs
over all elements of — <S.

240

GENERAL THEORY: ZERO-SUM n-PERSONS

The bilinear form K( { , y ) o( (17:2) in 17.4.1. is therefore

KU, v) - 2

T*,T-8

and finally

v(S) = Max— Min— K( £ , 77 ) = Min- Max— K( £ , rj ).

25.2. Discussion of the Concept

25 . 2 . 1 . The above function v(S) is defined for all subsets S of I and has
real numbers as values. Thus it is, in the sense of 13.1.3., a numerical
set function . We call it the characteristic function of the game r. As we
have repeatedly indicated, we expect to base the entire theory of the
zero-sum n-person game on this function.

It is well to visualize what this claim involves. We propose to determine
everything that can be said about coalitions between players, compensa-
tions between partners in every coalition, mergers or fights between coali-
tions, etc., in terms of the characteristic function v(S) alone. Prima facie ,
this program may seem unreasonable, particularly in view of these two facts:

(a) An altogether fictitious two-person game, which is related to the
real n-person game only by a theoretical construction, was used to define
v(S). Thus v(S) is based on a hypothetical situation, and not strictly
on the n-person game itself.

(b) v(S) describes what a given coalition of players (specifically,
the set S ) can obtain from their opponents (the set — S ) — but it fails to
describe how the proceeds of the enterprise are to be divided among the
partners k belonging to S. This division, the “ imputation,” is indeed
directly determined by the individual functions 3Ck(r h • • • , r n ), k belong-
ing to S y while v(S) depends on much less. Indeed, v(S) is determined by
their partial sum JC(r 5 , t~ s ) alone, and even by less than that since it is

the saddle value of the bilinear form K( £ , 77 ) based on 5C(r 5 , r ~ s ) (cf. the
formulae of 25.1.3.).

25 . 2 . 2 . In spite of these considerations we expect to find that the
characteristic function v(S) determines everything, including the “impu-
tation” (cf. (b) above). The analysis of the zero-sum three-person game in
Chapter V indicates that the direct distribution (i.e., “imputation”)
by means of the 3C*(ri, • • • , t„) is necessarily offset by some system of
“compensations” which the players must make to each other before coali-
tions can be formed. The “compensations” should be determined essen-
tially by the possibilities which exist for each partner in the coalition S
(i.e. for each k belonging to £), to forsake it and to join some other coalition
T. (One may have to consider also the influence of possible simultaneous
and concerted desertions by sets of several partners in S etc.) I.e. the
“imputation” of v(S) to the players k belonging to S should be determined

THE CHARACTERISTIC FUNCTION

241

by the other v(T) 1 — and not by the 3C*(ri, • * • , r n ). We have demon-
strated this for the zero-sum three-person game in Chapter V. One of the
main objectives of the theory we are trying to build up is to establish the
same thing for the general n-person game.

25.3. Fundamental Properties

25 . 3 . 1 . Before we undertake to elucidate the importance of the char-
acteristic function v(S) for the general theory of games, we shall investigate
this function as a mathematical entity in itself. We know that it is a
numerical set function, defined for all subsets S of I = (1, 2, • • • , n)
and we now propose to determine its essential properties.

It will turn out that they are the following:

(25 :3:a) v(©) = 0,

(25:3 :b) v(-S) = -v(S),

(25:3:c) v(S u T) * v(S) + v(T), if S n T = ©.

We prove first that the characteristic set function v(S) of every game
fulfills (25:3:a)-(25:3:c).

25 . 3 . 2 . The simplest proof is a conceptual one, which can be carried out
with practically no mathematical formulae. However, since we gave
exact mathematical expressions for v(S) in 25.1.3., one might desire a
strictly mathematical, formalistic proof — in terms of the operations Max
and Min and the appropriate vectorial variables. We emphasize therefore
that our conceptual proof is strictly equivalent to the desired formalistic,
mathematical one, and that the translation can be carried out without
any real difficulty. But since the conceptual proof makes the essential
ideas clearer, and in a briefer and simpler way, while the formalistic proof
would involve a certain amount of cumbersome notations, we prefer to
give the former. The reader who is interested may find it a good exercise
to construct the formalistic proof by translating our conceptual one.

25 . 3 . 3 . Proof of (25:3:a): 2 The coalition © has no members, so it always
gets the amount zero, therefore v(©) = 0.

Proof of (25:3:b): v(*S) and v( — S) originate from the same (fictitious)
zero-sum two-person game, — the one played by the coalition S against

1 All this is very much in the sense of the remarks in 4.3.3. on the role of “virtual”
existence.

1 Observe that we are treating even the empty set © as a coalition. The reader should
think this over carefully. In spite of its strange appearance, the step is harmless— and
quite in the spirit of general set theory. Indeed, it would be technically quite a nuisance
to exclude the empty set from consideration.

Of course this empty coalition has no moves, no variables, no influence, no gains,
and no losses. But this is immaterial.

The complementary set of ©, the set of all players 7, will also be treated as a possible
coalition. This too is the convenient procedure from the set-theoretical point of view.
To a lesser extent this coalition also may appear to be strange, since it has no opponents.
Although it has an abundance of members — and hence of moves and variables it will
(in a zero-sum game) equally have nothing to influence, and no gains or losses. But this
too is immaterial.

242

GENERAL THEORY: ZERO-SUM n-PERSONS

the coalition —S. The value of a play of this game for its two composite
players is indeed v(S) and v( — £) respectively. Therefore v( — £) = —v(S).

Proof of (25:3:c): The coalition S can obtain from its opponents (by
using an appropriate mixed strategy) the amount v(S) and no more. The
coalition T can obtain similarly the amount v(T) and no more. Hence the
coalition SuT can obtain from its opponents the amount v(S) + v(T),
even if the subcoalitions S and T fail to cooperate with each other. 1 Since
the maximum which the coalition S u T can obtain under any condition is
v(S u T) this implies v(S u T) ^ v(S) + v(T). 2

25.4. Immediate Mathematical Consequences

25.4.1. Before we go further let us draw some conclusions from the above
(25:3:a)-(25:3:c). These will be derived in the sense that they hold for
any numerical set function v(S) which fulfills (25:3:a)-(25:3:c) irrespective
of whether or not it is the characteristic function of a zero-sum n- person
game T.

(25:4) v(J) = 0.

Proof:* By (25:3:a), (25:3:b), v(7) = v(-Q) = -v(©) = 0.

(25:5) v(Si u • • • u S p ) ^ v(S i) + • • * + v(S 9 )

if Si, * • • , S p are pairwise disjunct subsets of 7.

Proof: Immediately by repeated application of (25:3:c).

(25:6) v(Si) + • • • + v(S p ) ^ 0

if S h • • * , S p are a decomposition of 7, i.e. pairwise disjunct
subsets of 7 with the sum 7.

Proof: We have Si u • • • u S p = 7, hence v(£i u • • • u S p ) = 0 by
(25:4). Therefore (25:6) follows from (25:5).

25.4.2. While (25:4)-(25:6) are consequences of (25:3:a)-(25:3:c), they —
and even somewhat less — can replace (25:3:a)-(25:3:c) equivalently.
Precisely:

(25 :A) The conditions (25:3:a)-(25:3:c) are equivalent to the asser-

tion of (25:6) for the values p = 1, 2, 3 only; but (25:6) must
then be stated for p = 1, 2 with an = sign, and for p = 3 with
a g sign.

1 Observe that we are now using S n T * ©. If S and T had common elements, we
could not break up the coalition S U T into the subcoalitions S and T.

* This proof is very nearly a repetition of the proof ofa-f-b+c^Oin 22.3.2. One
could even deduce our (25:3:c) from that relation: Consider the decomposition of 7 into
the three disjunct subsets S, T t — (S U T). Treat the three corresponding (hypothetical)
absolute coalitions as the three players of the zero-sum three-person game into which this
transforms r. Then v(S ), v(T), v(S U T) correspond to the —a, — b, c loc. cit. ; hence
a + b + c £ 0 means — v(jS) — v(T) 4* v(£ U T) gt 0; i.e. v(S U T) ^ v(S) -f v(T).

*For a y(S) originating from a game, both (25:3:a) and (25:4) are conceptually
contained in the remark of footnote 2 on p. 241.

GIVEN CHARACTERISTIC FUNCTION

243

Proof: (25:6) for p = 2 with an = sign states v(<S) + v(— S) — 0
(we write S for Si, hence Si is —S ) ; i.e. v(— S) — —v(S) which is exactly
(25:3 :b).

(25:6) for p = 1 with an = sign states v(J) =0 (in this case Si must be
I ) — which is exactly (25:4). Owing to (25:3:b), this is exactly the same
as (25:3:a). (Cf. the above proof of (25:4).)

(25:6) for p = 3 with an g sign states v(S) + v(T) + v(-(Su T )) ^ 0
(we write S, T for Si, Si; hence S a is — (<S u T )), i.e.

— v( — OSu T)) k v(S) + v(T).

By (25:3:b) this becomes v(S u T) ^ v(S) + v(T) which is exactly (25:3:c).

So our assertions are equivalent precisely to the conjunction of (25:3:a)-
(25:3 :c).

26. Construction of a Game with a Given Characteristic Function
26.1. The Construction

26.1.1. We now prove the converse of 25.3. 1. : That for any numerical set
function v(S) which fulfills the conditions (25:3:a)-(25:3:c) there exists a
zero-sum n- person game T of which this v(S) is the characteristic function.

In order to avoid confusion it is better to denote the given numerical
set function which fulfills (25:3:a)-(25:3:c) by v 0 (S). We shall define
with its help a certain zero-sum n- person game T, and denote the char-
acteristic function of this T by v(£). It will then be necessary to prove that
v(aS) s v 0 OS).

Let therefore a numerical set function v 0 (S) which fulfills (25:3:a)-
(25:3:c) be given. We define the zero-sum n- person game T as follows: 1

Each player k — 1, 2, • * • , n will, by a personal move, choose a subset
Sk of / which contains k . Each one makes his choice independently of the
choice of the other players. 2

After this the payments to be made are determined as follows:

Any set S of players, for which
(26:1) Sk = S for every k belonging to S

is called a ring. 8 * 4 Any two rings with a common element are identical. 6

1 This game T is essentially a more general analogue of the simple majority game of
three persons, defined in 21.1. We shall accompany the text which follows with footnotes
pointing out the details of this analogy.

2 The w-element set I has 2 n “ l subsets S containing k, which we can enumerate by an
index n(S) = 1, 2, • • • , 2"” 1 . If we now let the player k choose, instead of Sk , its
index r* — r*0S*) — 1, 2, • • • , 2 n ~ l , then the game is already in the normalized form of
11.2.3. Clearly all 0* ** 2 n_1 .

* The rings are the analogues of the couples in 21.1. The contents of footnote 3 on
p. 222 apply accordingly; in particular the rings are the formal concept in the set of rules
of the game which induces the coalitions which influence the actual course of each play.

4 Verbally: A ring is a set of players, in which every one has chosen just this set.

The analogy with the definition of a couple in 21.1. is clear. The differences are due
to formal convenience: in 21.1. we made each player designate the other element of the
couple which he desires; now we expect him to indicate the entire ring. A closer analysis
of this divergence would be easy enough, but it does not seem necessary.

6 Proof : Let S and T be two rings with a common element k; then by (26:1) Sk * S
and Sk “ T t and so S ** T.

244

GENERAL THEORY: ZERO-SUM n-PERSONS

In other words: The totality of all rings (which have actually formed in a
play) is a system of pairwise disjunct subsets of J.

Each player who is contained in none of the rings thus defined forms by
himself a (one-element) set which is called a solo set . Thus the totality
of all rings and solo sets (which have actually formed in a play) is a decompo-
sition of J; i.e. a system of pairwise disjunct subsets of I with the sum J.
Denote these sets by C if • • • , C v and the respective numbers of their
elements by n h • • • , n p .

Consider now a player k. He belongs to precisely one of these sets
C i, • • • , C p say to C q . Then player k gets the amount

v

(26:2) J_ Vo(( 7 9 ) _ I V Vo( c r ).>

71 q 71

r-1

This completes the description of the game I\ We shall now show that
this T is a zero-sum n- person game and that it has the desired characteristic
function v 0 (S).

26.1.2. Proof of the zero-sum character: Consider one of the sets C q .
Each one of the n q players belonging to it gets the same amount, stated in
(26:2). Hence the players of C q together get the amount

v

(26:3) Vo (C 9 ) - ^v 0 (C r ).

r-1

In order to obtain the total amount which all players 1, • • • , n get, we
must sum the expression (26:3) over all sets C q , i.e. over all q = 1, • • • , p.
This sum is clearly

£ V„(C.) - £ Vo (C r ),

<2 = 1 r-1

i.e. zero. 2

Proof that the characteristic function is v 0 (£) : Denote the characteristic
function of T by v(S). Remember that (25:3:a)-(25:3:c) hold for v(S)
because it is a characteristic function, and for v 0 (S) by hypothesis. Conse-
quently (25:4)-(25:6) also hold for both v(S) and v 0 (aS).

We prove first that

(26:4) v(S) ^ v 0 (S) for all subsets S of I .

If S is empty, then both sides are zero by (25:3:a). So we may assume that
S is not empty. In this case a coalition of all players k belonging to S can

1 The course of the play, that is the choices Si, • • • , S n — or, in the sense of footnote 2
on p. 243, the choices n, • • • , r» — determine the C j, • • • J C„, and thus the expression
(26:2). Of course (26:2) is the3C*(ri, • • • , r n ) of the general theory.

p

2 Obviously n q = n.

A-i

INESSENTIAL AND ESSENTIAL GAMES

245

govern the choices of its Sk so as with certainty to make S a ring. It suf-
fices for every k in S to choose his Sk = S. Whatever the other players
(in — S ) do, S will thus be one of the sets (rings or solo sets) Cj, • • • , C p ,
say C q . Thus each k in C q = S gets the amount (26:2); hence the entire
coalition S gets the amount (26:3). Now we know that the system

Ci, • • • , C P

p

is a decomposition of /; hence by (25:6) v 0 (C r ) ^ 0. That is, the

r- 1

expression (26:3) is ^ v 0 (C fl ) = Vo(S). 1 In other words, the players belong-
ing to the coalition S can secure for themselves at least the amount v 0 (5)
irrespective of what the players in —S do. This means that v($) ^ v 0 (S);
i.e. (26:4).

Now we can establish the desired formula
(26:5) v(S) = VoGS).

Apply (26:4) to — S. Owing to (25:3:b) this means — v(S) ^ ~v 0 (S), i.e.
(26:6) v(5) g v 0 (S).

(26:4), (26:6) give together (26:5). 2

26.2. Summary

26.2. To sum up: in paragraphs 25.3.-26.1. we have obtained a com-
plete mathematical characterization of the characteristic functions v(S) of
all possible zero-sum n- person games r. If the surmise which we expressed
in 25.2.1. proves to be true, i.e. if we shall be able to base the entire theory
of the game on the global properties of the coalitions as expressed by v(S),
then our characterization of v(aS) has revealed the exact mathematical
substratum of the theory. Thus the characterization of v(S) and the func-
tional relations (25:3:a)-(25:3:c) are of fundamental importance.

We shall therefore undertake a first mathematical analysis of the mean-
ing and of the immediate properties of these relations. We call the func-
tions which satisfy them characteristic functions — even when they are viewed
in themselves, without reference to any game.

27. Strategic Equivalence. Inessential and Essential Games
27.1. Strategic Equivalence. The Reduced Form

27.1.1. Consider a zero-sum n-person game T with the characteristic
function v(S). Let also a system of numbers aj, • • • , a® be given. We

1 Observe that the expression (26:3), i.e. the total amount obtained by the coalition S,
is not determined by the choices of the players in S alone. But we derived for it a lower
bound v 0 (&), which is determined.

* Observe that in our discussion of the good strategies of the (fictitious) two-person
game between the coalitions S and —S (our above proof really amounted to that), we
considered only pure strategies, and no mixed ones. In other words, all these two-person
games happened to be strictly determined.

This, however, is irrelevant for the end which we are now pursuing.

246

GENERAL THEORY: ZERO-SUM n-PERSONS

now form a new game r' which agrees with T in all details except for this:
T' is played in exactly the same way as T, but when all is over, player k gets
in T' the amount which he would have got in r (after the same play), plus a£.
(Observe that the a®, * • • , a® are absolute constants!) Thus if r is
brought into the normalized form of 11.2.3. with the functions

3C*(n, • • • , t„),

then T' is also in this normalized form, with the corresponding functions
3 C£(ti, * * • i t w ) = 3Cjk(ri, • • • , r n ) + a®. Clearly T' will be a zero-sum
n- person game (along with T) if and only if

(27:1) £ «2 = 0,

fc -1

which we assume.

Denote the characteristic function of T' by v'(S), then clearly
(27:2) v'(S) ^ v(5) + X «2-‘

fc in S

Now it is apparent that the strategic possibilities of the two games T and T'
are exactly the same. The only difference between these two games con-
sists of the fixed payments a% after each play. And these payments are
absolutely fixed; nothing that any or all of the players can do will modify
them. One could also say that the position of each player has been shifted
by a fixed amount, but that the strategic possibilities, the inducements and
possibilities to form coalitions etc., are entirely unaffected. In other words:
If two characteristic functions v(S) and v'(S) are related to each other by
(27 :2) 1 2 , then every game with the characteristic function v(S) is fully equiva-
lent from all strategic points of view to some game with the characteristic
function v'($), and conversely. I.e. v(S ) and v'(S) describe two strategi-
cally equivalent families of games. In this sense v(S ) and v'(S) may them-
selves be considered equivalent.

Observe that all this is independent of the surmise restated in 26.2.,
according to which all games with the same v(S) have the same strategic
characteristics.

27 . 1 . 2 . The transformation (27:2) (we need pay no attention to (27:1),
cf. footnote 2 above) replaces, as we have seen, the set functions v(>S) by a

1 The truth of this relation becomes apparent if one recalls how v(S), v'(S) were
defined with the help of the coalition S. It is also easy to prove (27:2) formalistically
with the help of the3C*(ri, • • • , r n ), 3C*(n, • • • , r n ).

‘Under these conditions (27:1) follows and need not be postulated separately.
Indeed, by (25:4) in 25.4.1., v(7) « v'(7) « 0, hence (27:2) gives

n

X at - 0; i.e. X “IE ”

kin I k — l

INESSENTIAL AND ESSENTIAL GAMES

247

strategically fully equivalent set-function v'(S). We therefore call this
relationship strategic equivalence .

We now turn to a mathematical property of this concept of strategic
equivalence of characteristic functions.

It is desirable to pick from each family of characteristic functions v(S)
in strategic equivalence a particularly simple representative v(S). The
idea is that, given v(S)> this representative v(S ) should be easy to determine,
and that on the other hand two v(S) and v'(S) would be in strategic equiva-
lence if and only if their representatives v(S) and v'(S ) are identical.
Besides, we may try to choose these representatives v(S) in such a fashion
that their analysis is simpler than that of the original v(S).

27 . 1 . 3 . When we started from characteristic functions v(S) and v'(S),
then the concept of strategic equivalence could be based upon (27:2) alone;
(27:1) ensued (cf. footnote 2, p. 246). However, we propose to start now
from one characteristic function v(S) alone, and to survey all possible v'(S )
which are in strategic equivalence with it — in order to choose the representa-
tive v(/S) from among them. Therefore the question arises which systems
a\ y • • • , a° we may use, i.e. for which of these systems (using (27:2)) the
fact that v(S) is a characteristic function entails the same for v'(aS). The
answer is immediate, both by what we have said so far, and by direct
verification: The condition (27:1) is necessary and sufficient. 1

Thus we have the n indeterminate quantities aj, • • • , at our
disposal in the search for a representative v{S); but the a\ y • • * , a® are
subject to one restriction: (27:1). So we have n — 1 free parameters at
our disposal.

27 . 1 . 4 . We may therefore expect that we can subject the desired repre-
sentative v(S) to w - 1 requirements. As such we choose the equations

(27:3) v((l)) = v((2)) = • • • = v((n)).>

I.e. we require that every one-man coalition — every player left to himself —
should have the same value.

We may substitute (27 :2) into (27 :3) and state this together with (27 :1),
and so formulate all our requirements concerning the aj, • * • , aj. So
we obtain:

(27:1*) £«*-<>,

Jfc-1

(27:2*) v((l)) + a? = v((2)) + «!-•••- v((»)) + «•.

It is easy to verify that these equations are solved by precisely one system of

/V 0 ... /-y 0 •

OL\y , (X n .

1 This detailed discussion may seem pedantic. We gave it only to make clear that
when we start with two characteristic functions v(S) and v'(<S) then (27:1) is superfluous,
but when we start with one characteristic function only, then (27:1) is needed.

* Observe that these are n — 1 and not n equations.

248

GENERAL THEORY: ZERO-SUM n-PERSONS

(27:4) «{=-v((4)) + ^v((j)).'

;-i

So we can say:

(27: A) We call a characteristic function v(S) reduced if and only if it

satisfies (27:3). Then every characteristic function v(S) is in
strategic equivalence with precisely one reduced v(S). This
v(S) is given by the formulae (27 :2) and (27 :4), and we call it the
reduced form of v(S).

The reduced functions will be the representatives for which we have
been looking.

27.2. Inequalities. The Quantity r

27.2. Let us consider a reduced characteristic function v(*S). We
denote the joint value of the n terms in (27 :3) by — 7 , i.e.

(27:5) - 7 = v((l)) = v((2)) = • • • v((n)).

We can state (27:5) also this way:

(27:5*) v(S) = —7 for every one-element set S.

Combination with (25:3:b) in 25.3.1. transforms (27:5*) into

(27:5**) v(S) = 7 for every (n — l)-element set S.

We re-emphasize that any one of (27:5), (27:5*), (27:5**) is — besides
defining 7 — just a restatement of (27 :3), i.e. a characterization of the reduced
nature of v(S).

Now apply (25:6) in 25.4.1. to the one-element sets Si = ( 1 ), • • • ,
S n = ( n ). (So p = n). Then (27:5) gives —ny g 0 , i.e.:

(27:6) 7 = 0 .

Consider next an arbitrary subset S of /. Let p be the number of its
elements: S = (k if • • • , k p ). Now apply (25:5) in 25.4.1. to the one-
element sets Si = (&i), ••*,£„= (kp). Then (27:5) gives

vOS) £ - py .

Apply this also to — S which has n — p elements. Owing to (25:3:b) in
25.3.1., the above inequality now becomes

—?(S) ^ — (n — p)y; i.e. v(S) g (n — p)y.

1 Proof: Denote the joint value of the n terms in (27 :2*) by 0. Then (27:2 ,,, ) amounts
to al * — v((fc)) + 0, and so (27:1*) becomes

n 1 n

n/3 - v ((*0) “ 0; i.e. /3 - - ^ v((A;)).

k-l n h - 1

INESSENTIAL AND ESSENTIAL GAMES

249

Combining these two inequalities gives :

(27:7) — py ^ v(S) g (n — p)y for every p-element set S.

(27:5*) and ^(©) = 0 (i.e. (25:3:a) in 25.3.1.) can also be formulated
this way:

(27:7*) For p = 0, 1 we have = in the first relation of (27:7).

(27:5**) and v(7) = 0 (i.e. (25:4) in 25.4.1.) can also be formulated
this way:

(27:7**) For p = n - 1, n we have = in the second relation of (27:7).

27.3. Inessential ity and Essentiality

27 . 3 . 1 . In analyzing these inequalities it is best now to distinguish two
alternatives.

This distinction is based on (27 :6) :

First case: 7 = 0. Then (27:7) gives v(£) = 0 for all S. This is a
perfectly trivial case, in which the game is manifestly devoid of further
possibilities. There is no occasion for any strategy of coalitions, no element
of struggle or competition: each player may play a lone hand, since there
is no advantage in any coalition. Indeed, every player can get the amount
zero for himself irrespective of what the others are doing. And in no
coalition can all its members together get more than zero. Thus the value
of a play of this game is zero for every player, in an absolutely unequivocal
way.

If a general characteristic function v(S) is in strategic equivalence
with such a v(<S) — i.e. if its reduced form is v(S) = 0 — then we have the
same conditions, only shifted by ag for the player k. A play of a game T
with this characteristic function v(S) has unequivocally the value a \ g for
the player k : he can get this amount even alone, irrespective of what the
others are doing. No coalition could do better in toto.

We call a game T, the characteristic function v(S) of which has such a
reduced form v(S) = 0 , inessential. 1

27 . 3 . 2 . Second case: 7 > 0. By a change in unit 2 we could make 7 = l . 3
This obviously affects none of the strategically significant aspects of the
game, and it is occasionally quite convenient to do. At this moment, how-
ever, we do not propose to do this.

In the present case, at any rate, the players will have good reasons to
want to form coalitions. Any player who is left to himself loses the amount
7 (i.e. he gets — 7 , cf. (27:5*) or (27:7*)), while any n — 1 players who

1 That this coincides with the meaning given to the word inessential in 23.1.3. (in the
special case of a zero-sum three-person game) will be seen at the end of 27.4.1.

a Since payments are made, we mean the monetary unit. In a wider sense it might
be the unit of utility. Cf. 2.1.1.

3 This would not have been possible in the first case, where 7*0.

250

GENERAL THEORY: ZERO-SUM n-PERSONS

cooperate win together the amount 7 (i.e. their coalition gets 7 , cf. (27:5**)
or (27:7**)).'

Hence an appropriate strategy of coalitions is now of great importance.
We call a game T essential when its characteristic function v(S) has a
reduced form v(S) not = 0. 1 2 *

27.4. Various Criteria. Non-additive Utilities

27 . 4 . 1 . Given a characteristic function v(S), we wish to have an explicit
expression for the 7 of its reduced form v(S). (Cf. above.)

Now —7 is the joint value of the v((fc)), i.e. of the v((fc)) + and this

n

is by (27:4) - ^ v((j )). 8 Hence
j- 1

n

(27:8) 7 = ~ \ ^ v((j)).

1

Consequently we have:

(27 :B) The game T is inessential if and only if

Z v (0')) = 0 0-e- 7 = 0),

and it is essential if and only if

Z v (0’)) < 0 (i-e. 7 > 0). 4 *

j-i

For a zero-sum three-person game we have, with the notations of 23.1.,
v((l)) = —a, v((2)) = — 6 , v((3)) = — c; so 7 = £A. Therefore our con-
cepts of essential and inessential specialize to those of 23.1.3. in the case
of a zero-sum three-person game. Considering the interpretation of these
concepts in both cases, this was to be expected.

1 This is, of course, not the whole story There may be other coalitions — of > 1
but < n — 1 players — which are worth aspiring to. (If this is to happen, n — 1 must
exceed 1 by more than 1 , — i.e. n ^ 4.) This depends upon the V($) of the sets S with
> 1 but < n — 1 elements. But only a complete and detailed theory of games can
appraise the role of these coalitions correctly.

Our above comparison of isolated players and n — 1 player coalitions (the biggest
coalitions which have anybody to oppose!) suffices only for our present purpose: to
establish the importance of coalitions in this situation.

* Cf. again footnote 1 on p. 249.

* So —7 is the 0 of footnote 1 on p. 248.

n

4 We have seen already that one or the other must be the case, since 2) v((/)) ^0 as

i-i

well as 7 £ 0.

INESSENTIAL AND ESSENTIAL GAMES

251

27.4.2. We can formulate, some other criteria of inessentiality:
(27 :C) The game T is inessential if and only if its characteristic function

v(S) can be given this form :

v(S) s £ oj

*in 8

for a suitable system a?, * * * ,

Proof: Indeed, this expresses by (27:2) precisely that v(S) is in strategic
equivalence with v(S ) = 0. As this v(S ) is reduced, it is then the reduced
form of v(S) — and this is the meaning of inessentiality.

(27 :D) The game T is inessential if and only if its -characteristic

function v(S) has always = in (25:3:c) of 25.3.1.; i.e. when

v(S u T) = v(S) + v(T) if SnT = ©.

Proof: Necessity: A v{S) of the form given in (27 :C) above obviously
possesses this property.

Sufficiency: Repeated application of this equation gives = in (25:5) of
25.4.1.; i.e.

vOSi U • • • u Sp) = v(Si) + • • • + v(> Sp)

if Si, • • • , S p are pairwise disjunct.

Consider an arbitrary S, say S = (k h • • • , k p ). Then S i = {k i), • • • ,
S P = (k p ) give

v(fl) = v((40) + • • • + v(»,)).

So we have

v(S) = l al

kin 8

with aj = v((l)), • • • , a® = v((n)) and so r is inessential by (27 :C).

27.4.3. Both criteria (27 :C) and (27 :D) express that the values of all
coalitions arise additively from those of their constituents. 1 It will be
remembered what role the additivity of value, or rather its frequent absence,
has played in economic literature. The cases in which value is not gen-
erally additive were among the most important, but they offered sig-
nificant difficulties to every theoretical approach; and one cannot say
that these difficulties have ever been really overcome. In this connection
one should recall the discussions of concepts like complementarity, total
value, imputation, etc. We are now getting into the corresponding phase
of our theory; and it is significant that we find additivity only in the unin-

c

1 The reader will understand that we are using the word 11 value ” (of the coalition S)
for the quantity v($).

252

GENERAL THEORY: ZERO-SUM n-PERSONS

teresting (inessential) case, while the really significant (essential) games
have a non-additive characteristic function. 1

Those readers who are familiar with the mathematical theory of measure
will make this further observation: the additive v(S) — i.e. the inessential
games — are exactly the measure functions of /, which give I the total
measure zero. Thus the general characteristic functions v(S) are a new
generalization of the concept of measure. These remarks are in a deeper
sense connected with the preceding ones concerning economic value. How-
ever, it would lead too far to pursue this subject further. 2

27.5. The Inequalities in the Essential Case

27 . 5 . 1 . Let us return to the inequalities of 27.2., in particular to (27 :7),
(27:7*), (27:7**). For y = 0 (inessential case) everything is trivially
clear. Assume therefore that y > 0 (essential case).

- ny

Figure 50.

Abscissa: p, number of elements of S. Dot at 0, — 7 , 7 , 0 or heavy line: Range of
possible values v(S) for the S with the corresponding p.

Now (27:7), (27:7*), (27:7**) set a range of possible values for v(S)
for every number p of elements in S. This range is pictured for each
p = 0, 1, 2, • • • , n — 2, n — 1, n in Figure 50.

We can add the following remarks:

27 . 5 . 2 . First: It will be observed that in an essential game — i.e. when
y > 0 — necessarily n ^ 3. Otherwise the formulae (27:7), (27:7*),
(27:7**) — or Figure 50, which expresses their content — lead to a conflict:
For n = 1 or 2 an (n — l)-element set S has 0 or 1 elements, hence its

1 We are, of course, concerned at this moment only with a particular aspect of the
subject: we are considering values of coalitions only — i.e. of concerted acts of behavior —
and not of economic goods or services. The reader will observe, however, that the spe-
cialization is not as far reaching as it may seem : goods and services stand really for the
economic act of their exchange — i.e. for a concerted act of behavior.

*The theory of measure reappears in another connection. Cf. 41.3.3.

INESSENTIAL AND ESSENTIAL GAMES 253

v(S) must on the one hand be 7, and on the other hand 0 or —7, which is
impossible. 1

Second: For the smallest possible number of participants in an essential
game, i.e. for n = 3, the formulae (27:7), (27:7*), (27:7**) — or Figure 50 —
determine Everything: they state the values of v(S) for 0, 1, n — 1, n-e lement
sets S; and for n = 3 the following are all possible element numbers: 0, 1,2,
3. (Cf. also a remark in footnote 1 on p. 250.) This is in harmony with
the fact which we found in 23.1.3., according to which there exists only
one type of essential zero-sum three-person games.

Third: For greater numbers of participants, i.e. for n ^ 4, the problem
assumes a new complexion. As formulae (27:7), (27:7*), (27:7**) — or
Figure 50 — show, the element number of p of the set S can now have
other values than 0,1 , n — 1, n. I.e. the interval

(27:9) 2 g p ^ n - 2

now becomes available. 2 It is in this interval that the above formulae no
longer determine a unique value of v(S); they set for it only the interval

(27:7) -P7 ^ v(S) ^ (n - p) 7,

the length of which is ny for every p (cf. again Figure 50).

27 . 5 . 3 . In this connection the question may be asked whether really
the entire interval (27 :7) is available, — i.e. whether it cannot be narrowed
further by some new, more elaborate considerations concerning v(S).
The answer is: No. It is actually possible to define for every n ^ 4 a
single game T p in which, for each p of (27:9), v(S) assumes both values
— py and (n — p ) 7 for suitable p-element sets S. It may suffice to mention
the subject here without further elaboration.

To sum up: The real ramifications of the theory of games appear only
when n ^ 4 is reached. (Cf. footnote 1 on p. 250, where the same idea
was expounded.)

27,6. Vector Operations on Characteristic Functions
27 . 6 . 1 . In concluding this section some remarks of a more formal nature
seem appropriate.

The conditions (25:3:a)-(25:3:c) in 25.3.1., which describe the charac-
teristic function v(&), have a certain vectorial character: they allow ana-
logues of the vector operations, defined in 16.2. 1., of scalar multiplication , and
of vector addition . More precisely:

Scalar multiplication: Given a constant t ^ 0 and a characteristic func-
tion v(S), then tv(S) = u(£) is also a characteristic function. Vector
addition: Given two characteristic functions v(aS), w(S); s then

1 Of course, in a zero-sum one-person game nothing happens at all, and for the zero-
sum two-person games we have a theory in which no coalitions appear. Hence the
inessentiality of all these cases is to be expected.

2 It has n — 3 elements; and this number is positive as soon as n 4.

3 Everything here must refer to the same n and to the same set of players

/ - (1, 2, • • • , n).

254 GENERAL THEORY: ZERO-SUM n-PERSONS

v(S) + w (S) « z(S) is also a characteristic function. The only difference
from the corresponding definitions of 16.2. is that we had to require t ^ 0. 1 * 2

27 . 6 . 2 . The two operations defined above allow immediate practical
interpretation:

Scalar multiplication: If t = 0, then this produces u(S) = 0, i.e. the
eventless game considered in 27.3.1. So we may assume t > 0. In this
case our operation amounts to a change of the unit of utility, namely to its
multiplication by the factor t.

Vector addition: This corresponds to the superposition of the games
corresponding to v(S) and to w (S). One would imagine that the same
players 1, 2, • • • , n are playing these two games simultaneously, but
independently. I.e., no move made in one game is supposed to influence
the other game, as far as the rules are concerned. In this case the charac-
teristic function of the combined game is clearly the sum of those of the two
constituent games. 3

27 . 6 . 3 . We do not propose to enter upon a systematic investigation of
these operations, i.e. of their influence upon the strategic situations in the
games which they affect. It may be useful, however, to make some remarks
on this subject — without attempting in any way to be exhaustive.

We observe first that combinations of the operations of scalar multiplica-
tion and vector addition also can now be interpreted directly. Thus the
characteristic function

(27:10) zOS) ^ tv(S) + sw(S)

belongs to the game which arises by superposition of the games of v(S) and
w(S) if their units of utility are first multiplied by t and $ respectively.

If s = 1 — t y then (27:10) corresponds to the formation of the center of
gravity in the sense of (16:A:c) in 16.2.1.

It will appear from the discussion in 35.3.4. (cf. in particular footnote 1
on p. 304 below) that even this seemingly elementary operation can have
very involved consequences as regards strategy.

We observe next that there are some cases where our operations have no
consequences in strategy.

First, the scalar multiplication by a t > 0 alone, being a mere change in
unit, has no such consequences.

indeed, t < 0 would upset (25:3:c) in 25.3.1. Note that a multiplication of the
original 3C*(n, • • • , r n ) with a t < 0 would be perfectly feasible. It is simplest to
consider a multiplication by t — —1, i.e. a change in sign. But a change of sign of the
3 C*(ti, • • , r n ) does not at all correspond to a change of sign of the v(S). This should be

clear by common sense, as a reversal of gains and losses modifies all strategic considera-
tions in a very involved way. (This reversal and some of its consequences are familiar
to chess players.) A formal corroboration of our assertion may be found by inspecting
the definitions of 25.1.3.

1 Vector spaces with this restriction of scalar multiplication are sometimes called
positive vector spaces. We do not need to enter upon their systematic theory.

* This should be intuitively obvious. An exact verification with the help of 25.1.3.
involves a somewhat cumbersome notation, but no real difficulties.

GROUPS, SYMMETRY AND FAIRNESS

255

Second — and this is of greater significance — the strategic equivalence
discussed in 27.1. is a superposition: we pass from the game of v(S) to the
strategically equivalent game of v'(S ) by superposing on the former an
inessential game. 1 (Cf. (27:1) and (27 :2) in 27.1.1. and, concerning inessen-
tiality, 27.3.1. and (27 :C) in 27.4.2.) We may express this in the following
way: we know that an inessential game is one in which coalitions play no
role. The superposition of such a game on another one does not disturb
strategic equivalence, i.e. it leaves the strategic structure of that game
unaffected.

28. Groups, Symmetry and Fairness
28.1. Permutations, Their Groups, and Their Effect on a Game

28.1.1. Let us now consider the role of symmetry, or more generally, the
effects of interchanging the players 1, • • • , n — or their numbers — in an
n-person game T. This will naturally be an extension of the corresponding
study made in 17.11. for the zero-sum two-person game.

This analysis begins with what is in the main a repetition of the steps
taken in 17.11. for n = 2. But since the interchanges of the symbols
1, • • • , n offer for a general n many more possibilities than for n = 2, it is
indicated that we should go about it somewhat more systematically.

Consider the n symbols 1, • • • , n. Form any permutation P of these
symbols. P is described by stating for every i = 1, • • • , n, into which i p
(also = 1, • • • , n), P carries it. So we write:

(28:1) P: i — > i p )

or by way of complete enumeration:

«**> P: (f: 2 -: : : : ;

Among the permutations some deserve special mention:

(28:A:a) The identity I n which leaves every i(— 1, • • • , n) un-

changed:

i — ► i^ n = i.

(28:A:b) Given two permutations P, Q, their product PQ , which

consists in carrying out first P and then Q:

i {pq = (ir)Q.

1 With the characteristic function w(<S) * ^ aj, then in our above notatibns

k in 8

v'(S) - v(S) + w (8)

1 Thus for n » 2, the interchange of the two elements 1, 2 is the permutation go-
The identity (cf. below) is ^ j’

256 GENERAL THEORY: ZERO-SUM n-PERSONS

The number of all possible permutations is the factorial of n r

n! = 1 • 2 • . . . * n,

and they form together the symmetric group of permutations 2„. Any
subsystem G of 2 n which fulfills these two conditions:

(28:A:a*) I n belongs to (?,

(28:A:b*) PQ belongs to G if P and Q do,

is a group of permutations. 1

A permutation P carries every subset S of I = (1, • • • , n) into another
subset S p . 2

28.1.2. After these general and preparatory remarks we now proceed
to apply their concepts to an arbitrary n- person game T.

Perform a permutation P on the symbols 1, • • • , n denoting the players
of T. I.e. denote the player k = 1, • • • , n by k p instead of fc; this trans-
forms the game T into another game T p . The replacement of T by T p must
make its influence felt in two respects: in the influence which each player
exercises on the course of the play, — i.e. in the index k of the variable t*
which each player chooses; and in the outcome of the play for him, — i.e. in
the index k of the function X * which expresses this. 8 So T p is again in the
normalized form, with functions 3 C£(ti, ••• , r n ), k = 1, • • • , n. In
expressing 5Cj(ri, • • • , r n ) by means of 3C*(ri, • • • , r„), we must remem-
ber: the player k in T had X *; now he is k p in r p , so he has3C£>. If we form
X fa with the variables ti, • • • , r n , then we express the outcome of the
game T p when the player whose designation in T p is k chooses r*. So the
player k in T who is k p in T p chooses t*p. So the variables in 3C* must be
T i p ) * * • , r n p. We have therefore:

(28:3) X p k p(j\ y * • • , r n ) = Xk(riP y • • • , r n p). 4<6

1 For the important and extensive theory of groups compare L. C. Mathewson:
Elementary Theory of Finite Groups, Boston 1930; W. Burnside: Theory of Groups of
Finite Order, 2nd Ed. Cambridge 1911; A. Speiser : Theorie der Gruppen von endlicher
Ordnung, 3rd Edit. Berlin 1937.

We shall not need any particular results or concepts of group theory, and mention
the above literature only for the use of the reader who may want to acquire a deeper
insight into that subject.

Although we do not wish to tie up our exposition with the intricacies of group theory,
we nevertheless introduced some of its basic termini for this reason: a real understanding
of the nature and structure of symmetry is not possible without some familiarity with
(at least) the elements of group theory. We want to prepare the reader who may want to
proceed in this direction, by using the correct terminology.

For a fuller exposition of the relationship between symmetry and group theory, cf.
H. Weyl: Symmetry, Journ. Washington Acad, of Sciences, Vol. XXVIII (1938), pp.
253ff.

* If S * (fci, • • , k p ), then S p - (*f , • • • , k p ).

3 Cf. the similar situation for n = 2 in footnote 1 on p. 109.

4 The reader will observe that the superscript P for the index k of the functions 3C
themselves appears on the left-hand side, while the superscript P for the indices k of the
variables r* appear on the right-hand side. This is the correct arrangement; and the
argument preceding (28:3) was needed to establish it.

The importance of getting this point faultless and clear lies in the fact that we could

GROUPS, SYMMETRY AND FAIRNESS

257

Denote the characteristic functions of T and T p by v(S) and v p (S)
respectively. Since the players, who form in T p the set S p , are the same ones
who form in T the set S , we have

(28:4) y p (S p ) = v(5) for every S. 1

28 . 1 . 3 . If (for a particular P) T coincides with T p , then we say that T is
invariant or symmetric with respect to P. By virtue of (28:3) this is
expressed by

(28:5) 3C**(ti, ‘ * * , r n ) = 3C*(tip, * • • , r*/>).

When this is the case, then (28:4) becomes
(28:6) v(S p ) 3= y(S) for every S.

Given any T, we can form the system G r of all P with respect to which T
is symmetric. It is clear from (28:A:a), (28:A:b) above, that the identity I n
belongs to G r, and that if P, Q belong to Gr, then their product PQ does too.
So Gr is a group by (28:A:a*), (28:A:b*) above. We callGr the invariance
group of T.

Observe that (28:6) can now be stated in this form:

(28:7) v(S) = v(T) if there exists a P in Gr with S p = T,

i.e. which carries S into T .

The size of Gr — i.e. the number of its elements — gives some sort of a
measure of “how symmetric” T is. If every permutation P (other than
identity J n ) changes T, then Gr consists of 7 n alone, — r is totally unsymmetric .
If no permutation P changes T, then G r contains all P, i.e. it is the sym-
metric group — r is totally symmetric. There are, of course, numerous
intermediate cases between these two extremes, and the precise structure
of r’s symmetry (or lack of it) is disclosed by the group Gr.

28 . 1 . 4 . The condition after (28:7) implies that S and T have the same
number of elements. The converse implication, however, need not be
true if Gr is small enough, i.e. if T is unsymmetric enough. It is therefore

not otherwise be sure that successive applications of the superscripts P and Q (in this
order) to r will give the same result as a (single) application of the superscript PQ to T.
The reader may find the verification of this a good exercise in handling the calculus of
permutations.

For n — 2 and P = (to , application of P on either side had the same effect, so it is

not necessary to be exhaustive on this point. Cf. footnote 1 on p. 109.

6 In the zero-sum two-person game, X e3Cj a — 3C 2 , and similarly 3C F m 3Cf = — .

Hence in this case ^cf . above, n — 2 and P — GO) (28:3) becomes 3C p (n ,n) = — 3C(r*,n).

This is in accord with the formulae of 14.6. and 17.11.2.

But this simplification is possible only in the zero-sum two-person game; in all
other cases we must rely upon the general formula (28:3) alone.

1 This conceptual proof is clearer and simpler than a computational one, which could
be based on the formulae of 25.1.3. The latter, however, would cause no difficulties
either, only more extensive notations.

258

GENERAL THEORY: ZERO-SUM n-PERSONS

of interest to consider those groups G = Gr which permit this converse
implication, i.e. for which the following is true:

(28:8) If S, T have the same number of elements, then there exists

a P in G with S p = T, — i.e. which carries S into T .

This condition (28 :8) is obviously satisfied when G is the symmetric group
i.e. for the G = Gr = 2 n of a totally symmetric T. It is also satisfied
for certain smaller groups, — i.e. for certain r of less than total symmetry. 1

28.2. Symmetry and Fairness

28 . 2 . 1 . At any rate, whenever (28:8) holds for G = Gr, we can conclude
from (28:7):

(28:9) v(S) depends only upon the number of elements in S.

That is:

(28:10) v(S) = v p

where p is the number of elements in S, (p = 0, 1, • • • , n).

Consider the conditions (25:3:a)-(25:3:c) in 25.3.1., which give an
exhaustive description of all characteristic functions v(£). It is easy to
rewrite them for v p when (28:10) holds. They become:

(28:11 :a) v 0 = 0,

(28:11 :b) v„_ p == -v p ,

(28:11 :c) v p+q ^ v p + v q for p + q ^ n.

(27:3) in 27.1.4. is clearly a consequence of (28:10) (i.e. of (28:9)),
so that such a v(aS) is automatically reduced, — with 7 = — Vi. We have
therefore, in particular, (27:7), (27:7*), (27:7**) in 27.2., i.e. the conditions
of Figure 50.

Condition (28:11 :c) can be rewritten, by a procedure which is parallel
to that of (25 :A) in 25.4.2.

1 For n = 2, X n contains, besides the identity, only one more permutation ( p

cf. several preceding references); so G = is the only possibility of any symmetry.

Consider therefore n ^ 3, and call G set-transitive if it fulfills (28:8). The question,
which G t** 2 n are then set-transitive, is of a certain group-theoretical interest, but we
need not concern ourselves with it in this work.

For the reader who is interested in group theory we nevertheless mention:

There exists a subgroup of 2» which contains half of its elements (i.e. in!), known as
the alternating group On. This group is of great importance in group theory and has been
extensively discussed there. For n ^ 3 it is easily seen to be set-transitive too.

So the real question is this: for which n <£ 3 do there exist set-transitive groups
G 5* 2„; CU?

It is easy to show that for n =* 3, 4 none exist. For n » 5, 6 such groups do exist.
(For n « 5 a set-transitive group G with 20 elements exists, while X5, Cts have 120, 60
elements respectively. For n * 6 a set-transitive group G with 120 elements exists,
while Sg, &6 have 720, 360 elements respectively.) For n =* 7, 8 rather elaborate group-
theoretical arguments show that no such groups exist. For n « 9 the question is still
open. It seems probable that no such groups exist for any n > 9, but this assertion has
not yet been established for all these n.

GROUPS, SYMMETRY AND FAIRNESS 269

Put r = n — p — g; then (28:11 :b) permits us to state (28:ll:c) as
follows:

(28:11 :c’ , ‘) v p + v fl + v r «£ 0 if p + q + r = n.

Now (28:11 :c*) is symmetric with respect to p, g, r; 1 hence we may make
V ^ q ^ r by an appropriate permutation. Furthermore, when p = 0
(hence r*= n — g), then (28:11 :c*) follows from (28:11 :a), (28:11 :b)
(even with = ). Thus we may assume p ^ 0. So we need to require
(28:11 :c*) only for 1 ^ p ^ g ^ r, and therefore the same is true for
(28:11 :c). Observe finally that, as r = n — p — g, the inequality g r
means p + 2g ^ n. We restate this:

(28:12) It suffices to require (28:11 :c) only when

1 ^ p ^ g, p + 2g ^ n. 2

28 . 2 . 2 . The property (28:10) of the characteristic function is a conse-
quence of symmetry, but this property is also important in its own right.
This becomes clear when we consider it in the simplest possible special case:
for n — 2.

Indeed, for n = 2 (28:10) simply means that the v' of 17.8.1. vanishes. 3
This means in the terminology of 17.11.2., that the game V is fair. We
extend this concept: The n-person game r is fair when its characteristic
function v(S) fulfills (28:9), i.e. when it is a v p of (28:10). Now, as in
17.11.2., this notion of fairness of the game embodies what is really essential
in the concept of symmetry. It must be remembered, however, that the
concept of fairness — and similarly that of total symmetry — of the game
may or may not imply that all individual players can expect the same fate
in an individual play (provided that they play well). For n = 2 this
implication did hold, but not for n ^ 3 ! (Cf. 17.11.2. for the former, and
footnotes 1 and 2 on p. 225 for the latter.)

28 . 2 . 3 . We observe, finally, that by (27:7), (27:7*), (27:7**) in 27.2., or
by Figure 50, all reduced games are symmetric and hence fair, when n = 3,
but not when n ^ 4. (Cf. the discussion in 27.5. 2.) Now the unrestricted
zero-sum n- person game is brought into its reduced form by the fixed
extra payments ai, • • • , a n (to the players 1, • • • , n, respectively), as
described in 27.1. Thus the unfairness of a zero-sum three-person game —
i.e, what is really effective in its asymmetry — is exhaustively expressed
by these a h a 2) a 3 ; that is, by fixed, definite payments. (Cf. also the
“basic values/ 7 a', 6', c' of 22.3.4.) In a zero-sum n- person game with

1 Both in its assertion and in its hypothesis!

* These inequalities replace the original p + q ^ n; they are obviously much stronger.

As they imply 3p ^ p + 2g ^ n and 1 + 2q ^ p + 2q ^ n, we have

n n — 1

P ^ £ ~ 2 ~'

1 By definition v' — v((l)) *■ — v((2)). For n - 2 the only essential assertion of
(28:9) (which is equivalent to (28:10)) is v((l)) *■ v((2)). Due to the above, this means
precisely that v # « — v', i.e. that v' - 0.

260

GENERAL THEORY: ZERO-SUM n-PERSONS

n ^ 4, this is no longer always possible, since the reduced form need
not be fair. That is, there may exist, in such a game, much more funda-
mental differences between the strategic positions of the players, which
cannot be expressed by the ai, • • • , a n > — i.e. by fixed, definite payments.
This will become amply clear in the course of Chapter VII. In the same
connection it is also useful to recall footnote 1 on p. 250.

29. Reconsideration of the Zero-sum Three-person Game
29.1. Qualitative Discussion

29.1.1. We are now prepared for the main undertaking: To formulate
the principles of the theory of the zero-sum n-person game. 1 The character-
istic function v(S), which we have defined in the preceding sections, provides
the necessary tool for this operation.

Our procedure will be the same as before: We must select a special case
to serve as a basis for further investigation. This shall be one which we
have already settled and which we nevertheless deem sufficiently charac-
teristic for the general case. By analyzing the (partial) solution found
in this special case, we shall then try to crystallize the rules which should
govern the general case. After what we said in 4.3.3. and in 25.2.2., it
ought to be plausible that the zero-sum three-person game will be the special
case in question.

29.1.2. Let us therefore reconsider the argument by which our present
solution of the zero-sum three-person game was obtained. Clearly the
essential case will be the one of interest. We know now that we may as
well consider it in its reduced form, and that we may also choose y — l. 2
The characteristic function in this case is completely determined, as dis-
cussed in the second case of 27.5.2.:

1 0 0

| when S has * elements. 8
1 A

0 3

We saw that in this game everything is decided by the (two-person)
coalitions which form, and our discussions 4 produced the following main
conclusions:

Three coalitions may form, and accordingly the three players will
finish the play with the following results:

1 Of course the general n-person game will still remain, but we shall be able to solve
it with the help of the zero-sum games. The greatest step is the present one : the passage
to the zero-sum n-person games.

* Cf. 27.1.4. and 27.3.2.

1 In the notation of 23.1.1. this means a — b — c — 1. The general parts of the
discussions referred to were those in 22.2., 22.3., 23. The above specialization takes us
actually back to the earlier (more special) case of 22.1. So our considerations of 27.1.
(on strategic equivalence and reduction) have actually this effect in the zero-sum three-
person games: they carry the general case back into the preceding special one, as stated
above.

4 In 22.2.2., 22.2.3.; but these are really just elaborations of those in 22.1.2., 22.1.3.

RECONSIDERATION

261

\] Player

Coa\
lition \

1

2

3

(1, 2)

i

i

-1

(1, 3)

§

-1

i

(2, 3)

-1

i

i

Figure 51.

This “ solution ” calls for interpretation, and the following remarks suggest
themselves in particular: 1

29.1.3.

The three distributions specified above correspond to all
strategic possibilities of the game.

None of them can be considered a solution in itself; it is the
system of all three and their relationship to each other which
really constitute the solution.

The three distributions possess together, in particular, a
“ stability ” to which we have referred thus far only very
sketchily. Indeed no equilibrium can be found outside of
these three distributions; and so one should expect that any
kind of negotiation between the players must always in fine
lead to one of these distributions.

Again it is conspicuous that this “stability ’ ’ is only a
characteristic of all three distributions viewed together. No
one of them possesses it alone; each one, taken by itself, could
be circumvented if a different coalition pattern should spread
to the necessary majority of players.

29.1.4. We now proceed to search for an exact formulation of the
heuristic principles which lead us to the solutions of Figure 51, always
keeping in mind the remarks (29:A:a)-(29:A:d).

A more precise statement of the intuitively recognizable “stability”
of the system of three distributions in Figure 51 — which should be a concise
summary of the -discussions referred to in footnote 4 on p. 260 — leads us
back to a position already taken in the earlier, qualitative discussions. 2
It can be put as follows:

(29:B:a) If any other scheme of distribution should be offered for

consideration to the three players, then it will meet with

1 These remarks take up again the considerations of 4.3.3. In connection with
(2®:A:d) the second half of 4.6.2- may also be recalled.

* These viewpoints permeate all of 4.4.-4.6., but they appear more specifically in
4.4.1. and 4.6.2.

(29:A:a)
(29 :A :b)

(29 :A :c)

(29:A:d)

262 GENERAL THEORY: ZERO-SUM n-PERSONS

rejection for the following reason: a sufficient number of
players 1 prefer, in their own interest, at least one of the dis-
tributions of the solution (i.e. of Figure 51), and are con-
vinced or can be convinced 2 of the possibility of obtaining the
advantages of that distribution.

(29:B:b) If, however, one of the distributions of the solution is

offered, then no such group of players can be found.

We proceed to discuss the merits of this heuristic principle in a more
exact way.

29.2. Quantitative Discussion

29.2.1. Suppose that ft, ft, ft is a possible method of distribution
between the players 1,2,3. I.e.

ft + ft + 03 = 0.

Then, since by definition v((i))( = — 1 ) is the amount that player i can get
for himself (irrespective of what all others do), he will certainly block any
distribution with ft < v((z)). We assume accordingly that

ft ^ v((i)) = - 1 .

We may permute the players 1,2,3 so that

0i ^ ft ^ ft.

Now assume ft < i . Then a fortiori ft < %. Consequently the players
2,3 will both prefer the last distribution of Figure 51, 3 where they both get
the higher amount 7. 4 Besides, it is clear that they can get the advantage of
that distribution (irrespective of what the third player does), since the
amounts i, £ which it assigns to them do not exceed together v((2, 3)) = 1.

If, on the other hand, ft ^ then a fortiori ft ^ Since ft ^ — 1,
this is possible only when ft = ft = i, ft = — 1 , i.e. when we have the first
distribution of Figure 51. (Cf. footnote 3 above.)

1 Of course, in this case, two.

* What this “convincing” means was discussed in 4.4.3. Our discussion which
follows will make it perfectly clear.

* Since we made an unspecified permutation of the players 1,2,3 the last distribution
of Fig. 51 really stands for all three.

4 Observe that each one of these two players profits by such a change separately and
individually. It would not suffice to have only the totality (of these two) profit. Cf.,
e.g., the first distribution of Fig. 51 with the second; the players 1,3 as a totality would
profit by the change from the former to the latter, — and nevertheless the first distribution
is just as good a constituent of the solution as any other.

In this particular change, player 3 would actually profit (getting J instead of — 1),
and for player 1 the change is indifferent (getting § in both cases). Nevertheless player 1
will not act unless further compensations are made — and these can be disregarded in this
connection. For a more careful discussion of this point, cf. the last part of this section.

EXACT FORM OF THE GENERAL DEFINITIONS 263

This establishes (29:B:a) at the end of 29.1.4. (29:B:b) loc. cit. is
immediate: in each of the three distributions of Figure 51 there is, to be
sure, one player who is desirous of improving his standing, 1 but since there is
only one, he is not able to do so. Neither of his two possible partners gains
anything by forsaking his present ally and joining the dissatisfied player:
already each gets i, and they can get no more in any alternative distribution
of Figure 51. 2

29.2.2. This point may be clarified further by some heuristic elaboration.

We see that the dissatisfied player finds no one who desires spontaneously
to be his partner, and he can offer no positive inducement to anyone to
join him ; certainly none by offering to concede more than £ from the proceeds
of their future coalition. The reason for regarding such an offer as ineffec-
tive can be expressed in two ways : on purely formal grounds this offer may
be excluded because it corresponds to a distribution which is outside the
scheme of Figure 51; the real subjective motive for which any prospective
partner would consider it unwise 3 to accept a coalition under such conditions
is most likely the fear of subsequent disadvantage, — there may be further
negotiations preceding the formation of a coalition, in which he would be
found in a particularly vulnerable position. (Cf. the analysis in 22.1.2.,
22.1.3.)

So there is no way for the dissatisfied player to overcome the indifference
of the two possible partners. We stress: there is, on the side of the two
possible partners no positive motive against a change into another distribu-
tion of Figure 51, but just the indifference characteristic of certain types of
stability. 4

30. The Exact Form of the General Definitions
30.1. The Definitions

30.1.1. We return to the case of the zero-sum n- person game T with
general n. Let the characteristic function of T be v(S).

We proceed to give the decisive definitions.

In accordance with the suggestions of the preceding paragraphs we
mean by a distribution or imputation a set of n numbers on, • • • , a n with
the following properties

(30:1) oti ^ v((i)) for i = 1, • • • , n,

n

(30:2) £ «* = 0.

»-l

1 The one who gets —1.

* The reader may find it a good exercise to repeat this discussion with a general (not
reduced) v(S), — i.e. with general o, 6, c, and the quantities of 22.3.4. The result is the
same; it cannot be otherwise, since our theory of strategic equivalence and reduction is
correct. (Cf. footnote 3 on p. 260.)

* Or unsound, or unethical.

4 At every change from one distribution of Fig. 51 to another, one player is definitely
against, one definitely for it; and so the remaining player blocks the change by his
indifference.

264 GENERAL THEORY: ZERO-SUM n-PERSONS

It may be convenient to view these systems «i, • • • , a* as vectors in the
n-dimensional linear space L n in the sense of 16.1 2.:

•••,«»(

• , n) is called effective for the imputa-
g v(S).

— ►

An imputation a dominates another imputation 0 , in symbols

a = {«i,

A set S (i.e. a subset of I = 1, • •
tion a , if

(30:3) £ «<

i in 5

« ** P ,

if there exists a set S with the following properties:

(30:4:a) S is not empty,

(30:4:b) S is effective for a ,

(30:4:c) a* > & for all i in S.

A set V of imputations is a solution if it possesses the following properties:

(30:5:a) No p in V is dominated by an a in V,

(30:5:b) Every ($ not in V is dominated by some a in V.

(30:5:a) and (30:5:b) can be stated as a single condition:

(30:5:c) The elements of V are precisely those imputations which

are undominated by any element of V.

(Cf. footnote 1 on p. 40.)

30 . 1 . 2 . The meaning of these definitions can, of course, be visualized
when we recall the considerations of the preceding paragraphs and also of
the earlier discussions of 4.4.3.

To begin with, our distributions or imputations correspond to the more
intuitive notions of the same name in the two places referred to. What we
call an effective set is nothing but the players who “are convinced or can be

convinced” of the possibility of obtaining what they are offered by a ; cf.
again 4.4.3 and (29:B:a) in 29.1.4. The condition (30:4:c) in the definition
of domination expresses that all these players have a positive motive for
— > — >

preferring a to p . It is therefore apparent that we have defined domi-
nation entirely in the spirit of 4.4.1., and of the preference described by
(29:B:a) in 29.1.4.

The definition of a solution agrees completely with that given in 4.5.3.,
as well as with (29:B:a), (29:B:b) in 29.1.4.

EXACT FORM OF THE GENERAL DEFINITIONS 265

30.2. Discussion and Recapitulation

30.2.1. The motivation for all these definitions has been given at the
places to which we referred in detail in the course of the last paragraph.
We shall nevertheless re-emphasize some of their nlain features — particu-
larly the concept of a solution.

We have already seen in 4.6. that our concept of a solution of a game
corresponds precisely to that of a “ standard of behavior” of everyday
parlance. Our conditions (30:5:a), (30:5:b), which correspond to the
conditions (4:A:a), (4:A:b) of 4.5.3., express just the kind of “inner sta-
bility” which is expected of workable standards of behavior. This was
elaborated further in 4.6. on a qualitative basis. We can now reformulate
those ideas in a rigorous way, considering the exact character which the
discussion has now assumed. The remarks we wish to make are these: 1

30.2.2.

(30:A:a)

Consider a solution V. We have not excluded for an
— ► — ►

imputation 0 in V the existence of an outside imputation a '

(not in V) with a'H |8. 2 If such an a ' exists, the attitude
of the players must be imagined like this: If the solution V
(i.e. this system of imputations) is “accepted” by the players 1,
• • * , n, then it must impress upon their minds the idea that

only the imputations 0 in V are “ sound ” ways of distribution.

An a ' not in V with a ' H 0 , although preferable to an
effective set of players, will fail to attract them, because it is
“unsound.” (Cf. our detailed discussion of the zero-sum three-
person game, especially as to the reason for the aversion of each
player to accept more than the determined amount in a coali-
tion. Cf . the end of 29.2. and its references.) The view of the

“unsoundness” of a ' may also be supported by the existence

of an a in V with a H a ' (cf. (30:A:b) below). All these
arguments are, of course, circular in a sense and again
depend on the selection of V as a “standard of behavior,” i.e.
as a criterion of “soundness.” But this sort of circularity is
not unfamiliar in everyday considerations dealing with
“soundness.”

1 The remarks (30:A:a)-(30:A:d) which follow are a more elaborate and precise
presentation of the ideas of 4.6.2. Remark (30:A:e) bears the same relationship to
4.6.3.

* Indeed, we shall see in (31 :M) of 31.2.3. that an imputation 0 , for which never
— > — ►

a ' h p f exists only in inessential games.

266

GENERAL THEORY: ZERO-SUM n-PERSONS

(30:A:d)

(30:A:b) If the players 1, • • • , n have accepted the solution V as a

“ standard of behavior/ 1 then the ability to discredit with the
help of V (i.e. of its elements) any imputation not in V, is
necessary in order to maintain their faith in V. Indeed, for

every outside a ' (not in V) there must exist an a in V with

a H a '. (This was our postulate (30:5:b).)

(30:A:c) Finally there must be no inner contradiction in V, i.e. for

a , in V, never a H p . (This was our other postulate
(30:5:a).)

(30:A:d) Observe that if domination, i.e. the relation fr* , were transi-

tive, then the requirements (30:A:b) and (30:A:c) (i.e. our
postulates (30:5:a) and (30:5:b)) would exclude the rather
delicate situation in (30:A:a). Specifically: In the situation of

(30:A:a), p belongs to V, a ' does not, and a'H By

(30:A:b) there exists an a in V so that a H a '. Now if

domination were transitive we could conclude that a H ft ,

which contradicts (30:A:c) since a , p both belong to V.
(30:A:e) The above considerations make it even more clear that only

V in its entirety is a solution and possesses any kind of stability
— but none of its elements individually. The circular char-
acter stressed in (30:A:a) makes it plausible also that several
solutions V may exist for the same game. I.e. several stable
standards of behavior may exist for the same factual situation.
Each of these would, of course, be stable and consistent in itself,
but in conflict with all others. (Cf. also the end of 4.6.3. and
the end of 4.7.)

In many subsequent discussions we shall see that this multiplicity of
solutions is, indeed, a very general phenomenon.

(30:A:e)

SO. 3 . The Concept of Saturation

30 . 3 . 1 . It seems appropriate to insert at this point some remarks of a
more formalistic nature. So far we have paid attention mainly to the
meaning and motivation of the concepts which we have introduced, but the
notion of solution, as defined above, possesses some formal features which
deserve attention.

The formal — logical — considerations which follow will be of no imme-
diate use, and we shall not dwell upon them extensively, continuing after-
wards more in the vein of the preceding treatment. Nevertheless we deem
that these remarks are useful here for a more complete understanding of the
structure of our theory. Furthermore, the procedures to be used here will
have an important technical application in an entirely different connection
in 51.1.-51.4.

EXACT FORM OF THE GENERAL DEFINITIONS 267

30 . 3 . 2 . Consider a domain (set) D for the elements x , y of which a certain
relation x(Ry exists. The validity of (R between two elements x, y of D is
expressed by the formula x(R y. 1 (R is defined by a statement specifying
unambiguously for which pairs x, y of D, x(R y is true, and for which it is not.
If x(R y is equivalent to y( Rx, then we say that x(Ry is symmetric. For any
relation (R we can define a new relation (R s by specifying x(R s y to mean the
conjunction of x(R y and y(Rx. Clearly (R 5 is always symmetric and coincides
with (R if and only if (R is symmetric. We call (R a the symmetrized form of (R . 2

We now define:

(30:B:a) A subset A of D is (R-satisfactory if and only if x(Ry holds

for all x, y of A.

(30:B:b) A subset A of D and an element y of D are (R-compatible

if and only if x(Ry holds for all x of A.

From these one concludes immediately:

(30:C:a) A subset A of D is (R-satisfactory if and only if this is true:

The y which are (R-compatible with A form a superset of A.

We define next :

(30:C:b) A subset A of D is (R-saturated if and only if this is true:

The y which are (R-compatible with A form precisely the set A.

Thus the requirement which must be added to (30 :C :a) in order to secure
(30:C:b) is this:

(30 :D) If y is not in A, then it is not (R-compatible with A) i.e.

there exists an x in A such that not x(Ry.

Consequently (R-saturation may be equivalently defined by (30:B:a) and
(30 :D).

30 . 3 . 3 . Before we investigate these concepts any further, we give some
examples. The verification of the assertions made in them is easy and will
be left to the reader.

First: Let D be any set and x(Ry the relation x = y. Then (R-satis-
factoriness of A means that A is either empty or a one-element set, while
(R-saturation of A means that A is a one-element set.

Second: Let D be a set of real numbers and x(Ry the relation x ^ y. z
Then (R-satisfactoriness of A means the same thing as above, 4 while (R-sat-
uration of A means that A is a one-element set, consisting of the greatest
element of D. Thus there exists no such A if D has no greatest element

1 It is sometimes more convenient to use a formula of the form <R(x, y), but for our
purposes x(Ry is preferable.

2 Some examples: Let D consist of ail real numbers. The relations x * y and x ^ y
are symmetric. None of the four relations x^y, x^y, x<y,x>y\B symmetric.
The symmetrized form of the two former is x ** y (conjunction of x £ y and x ^ y ),
the symmetrized form of the two latter is an absurdity (conjunction of x < y and x > y).

8 D could be any other set in which such a relation is defined, cf. the second example in
65.4.1.

4 Cf. footnote 1 on p. 268.

268 GENERAL THEORY: ZERO-SUM n-PERSONS

(e.g. for the set of all real numbers) and A is unique if D has a greatest
element (e.g. when it is finite).

Third: Let D be the plane and xGiy express that the points x , y have the
same height (ordinate). Then (R-satisfactoriness of A means that all
points of A have the same height, i.e. lie on one parallel to the axis of
abscissae. (R-saturation means that A is precisely a line parallel to the axis
of abscissae.

Fourth: Let D be the set of all imputations, and xGiy the negation of the
domination x h y. Then comparison of our (30:B:a), (30:D) with (30:5:a),
(30:5:b) in 30.1.1., or equally of (30:C:b) with (30:5:c) id. shows: (R-satura-
tion of A means that A is a solution.

30.3.4. One look at the condition (30:B:a) suffices to see that satis-
factoriness for the relation xGiy is the same as for the relation yGix and so
also for their conjunction x(R s y. In other words: (R-satisfactoriness is the
same thing as (R s -satisfactoriness.

Thus satisfactoriness is a concept which need be studied only on sym-
metric relations.

This is due to the x, y symmetric form of the definitory condition (30 :B :a) .
The equivalent condition (30:C:a) does not exhibit this symmetry, but of
course this does not invalidate the proof.

Now the definitory condition (30:C:b) for (R-saturation is very similar
in structure to (30:C:a). It is equally asymmetric. However, while
(30:C:a) possesses an equivalent symmetric form (30:B:a), this is not the
case for (30:C:b). The corresponding equivalent form for (30:C:b) is, as
we know, the conjunction of (30:B:a) and (30 :D) — and (30 :D) is not at all
symmetric. I.e. (30 :D) is essentially altered if x(R y is replaced by yC Rx.
So we see:

(30 :E) While (R-satisfactoriness in unaffected by the replacement of

(R by (R 5 , it does not appear that this is the case for (R-saturation.

Condition (30:B:a) (amounting to (R-satisfactoriness) is the same for
(R and (R 5 . Condition (30 :D) for (R s is implied by the same for (R since (R s
implies (R. So we see:

(30 :F) (R 5 -saturation is implied by (R-saturation.

The difference between these two types of saturation referred to above
is a real one: it is easy to give an explicit example of a set which is (R s -sat-
urated without being (R-saturated. 1

Thus the study of saturation cannot be restricted to symmetric relations.

30.3.6. For symmetric relations (R the nature of saturation is simple
enough. In order to avoid extraneous complications we assume for this
section that x(Rx is always true. 2

1 E.g.: The first two examples of 30.3.3. are in the relation of (R 5 and (R to each other
(cf. footnote 2 on p. 267); their concepts of satisfactoriness are identical, but those of
saturation differ.

* This is clearly the case for our decisive example of 30.3.3.: xGiy the negation of x H y
— since never x h x.

EXACT FORM OF THE GENERAL DEFINITIONS 269
Now we prove:

(30 :G) Let (R be symmetric. Then the (R-saturation of A is equiv-

alent to its being maximal (R-satisfactory. I.e. it is equivalent
to: A is (R-satisfactory, but no proper superset of A is.

Proof : (R-saturation means (R-satisfactoriness (i.e. condition (30:B:a))
together with condition (30 :D). So we need only prove: If A is ^-satis-
factory, then (30 :D) is equivalent to the non-(R-satisfactoriness of all proper
supersets of A.

Sufficiency of (30 :D): If B d A is (R-satisfactory, then any y in B, but
not in A f violates (30 :D). 1

Necessity of (30 :D): Consider a y which violates (30 :D). Then

B = A u (y) d A.

Now B is (R-satisfactory, i.e. for x', y' in B, always x'(Ry'. Indeed, when
x' y y ' are both in A, this follows from the (R-satisfactoriness of A. If x' f y'
are both = y, we are merely asserting yGiy. If one of x' f y' is in A, and the
other = 2 /, then the symmetry of (R allows us to assume x' in A, y' = y.
Now our assertion coincides with the negation of (30:D).

If (R is not symmetric, we can only assert this:

(30 :H) (R-saturation of A implies its being maximal (R-satisfactory.

Proof: Maximal (R-satisfactoriness is the same as maximal (R s -satisfactori-
ness, cf. (30 :E). As (R s is symmetric, this amounts to (R s -saturation by
(30 :G). And this is a consequence of (R-saturation by (30 :F).

The meaning of the result concerning a symmetric (R is the following:
Starting with any (R-satisfactory set, this set must be increased as long as
possible, — i.e. until any further increase would involve the loss of (R-satis-
factoriness. In this way in fine a maximal (R-satisfactory set is obtained,
— i.e. an (R-saturated one by (30:G). 2 This argument secures not only
the existence of (R-saturated sets, but it also permits us to infer that every
(R-satisfactory set can be extended to an (R-saturated one.

1 Note that none of the extra restrictions on (R has been used so far.

* This process of exhaustion is elementary — i.e. it is over after a finite number of
steps — when D is finite.

However, since the set of all imputations is usually infinite, the case of an infinite D
is important. When D is infinite, it is still heuristically plausible that the process of
exhaustion referred to can be carried out by making an infinite number of steps. This
process, known as transfinite induction } has been the object of extensive set-theoretical
studies. It can be performed in a rigorous way which is dependent upon the so-called
axiom of choice.

The reader who is interested will find literature in F. Hausdorfi \ footnote 1, on p. 61.
Cf. also E. ZermelOy Beweis dass jede Menge wohlgeordnet werden kann. Math. Ann.
Vol. 59 (1904) p. 514ff. and Math. Ann. Vol. 65 (1908) p. 107ff.

These matters carry far from our subject and are not strictly necessary for our
purposes. We do not therefore dwell further upon them.

270 GENERAL THEORY: ZERO-SUM n-PERSONS

It should be noted that every subset of an (R-saturated set is necessarily
(R-satisfactory . 1 The above assertion means therefore that the converse
statement is also true.

30 . 3 . 6 . It would be very convenient if the existence of solutions in our
theory could be established by such methods. The prima facie evidence,
however, is against this: the relation which we must use, x(R y — negation of
the domination x H y ) cf. 30.3.3. — is clearly asymmetrical. Hence we can-
not apply (30 :G), but only (30 :H): maximal satisfactoriness is only neces-
sary, but may not be sufficient for saturation, i.e. for being a solution.

That this difficulty is really deep seated can be seen as follows: If we could
replace the above (R by a symmetric one, this could not only be used to prove
the existence of solutions, but it would also prove in the same operation
the possibility of extending any (R-satisfactory set of imputations to a solution
(cf. above). Now it is probable that every game possesses a solution, but
we shall see that there exist games in which certain satisfactory sets are
subsets of no solutions. 2 * Thus the device of replacing (R by something
symmetric cannot work because this would be equally instrumental in prov-
ing the first assertion, which is presumably true, and the second one, which
is certainly not true. 8

The reader may feel that this discussion is futile, since the relation x(R y
which we must use (“not x H y”) is de facto asymmetric. From a technical
point of view, however, it is conceivable that another relation xgy may be
found with the following properties: zg y is not equivalent to x(Ry; indeed,
g is symmetric, while (R is not, butg-saturation is equivalent to (R-saturation.
In this case (R-saturated sets would have to exist because they are the
g-saturated ones, and the g-satisfactory — but not necessarily the ^-satis-
factory — sets would always be extensible to g-saturated, i.e. (R-saturated
ones. 4 * This program of attack on the existence problem of solutions is not
as arbitrary as it may seem. Indeed, we shall see later a similar problem
which is solved in precisely this way (cf. 51.4.3.). All this is, however, for
the time being just a hope and a possibility.

30 . 3 . 7 . In the last section we considered the question whether every
(R-satisfactory set is a subset of an (R-saturated set. We noted that for
the relation x(Ry which we must use (“not x H y” asymmetric) the answer
is in the negative. A brief comment upon this fact seems to be worth while.

If the answer had been in the affirmative it would have meant that any
set fulfilling (30:B:a) can be extended to one fulfilling (30:B:a) and(30:D);
or, in the notations of 30.1.1., that any set of imputations fulfilling (30:5:a)
can be extended to one fulfilling (30:5:a) and (30:5:b).

1 Clearly property (30:B:a) is not lost when passing to a subset.

* Cf. footnote 2 on p. 285.

8 This is a rather useful principle of the technical side of mathematics. The inappro-
priateness of a method can be inferred from the fact that if it were workable at all it
would prove too much.

4 The point is that (R- and 8-saturation are assumed to be equivalent to each other,

but (H- and S-satisfactoriness are not expected to be equivalent.

EXACT FORM OF THE GENERAL DEFINITIONS 271

It is instructive to restate this in the terminology of 4.6.2. Then the
statement becomes: Any standard of behavior which is free from inner
contradiction can be extended to one which is stable, — i.e. not only free
from inner contradictions, but also able to upset all imputations outside
of it.

The observation in 30.3.6., according to which the above is not true in
general, is of some significance: in order that a set of rules of behavior
should be the nucleus (i.e. a subset) of a stable standard of behavior, it
may have to possess deeper structural properties than mere freedom from
inner contradictions. 1

30.4. Three Immediate Objectives

30 . 4 . 1 . We have formulated the characteristics of a solution of an unre-
stricted zero-sum n- person game and can therefore begin the systematic
investigation of the properties of this concept. In conjunction with the
early stages of this investigation it seems appropriate to carry out three
special enquiries. These deal with the following special cases:

First: Throughout the discussions of 4., the idea recurred that the unso-
phisticated concept of a solution would be that of an imputation, — i.e. in
our present terminology, of a one-element set V. In 4.4.2. we saw specifi-
cally that this would amount to finding a “ first” element with respect to
domination. We saw in the subsequent parts of 4., as well as in our exact
discussions of 30.2., that it is mainly the intransitivity of our concept of
domination which defeats this endeavor and forces us to introduce sets of
imputations V as solutions.

It is, therefore, of interest — now that we are in a position to do it — to
give an exact answer to the following question: For which games do one-
element solutions V exist? What else can be said about the solutions of
such games?

Second: The postulates of 30.1.1. were extracted from our experiences
with the zero-sum three-person game, in its essential case. It is, therefore,
of interest to reconsider this case in the light of the present, exact theory.
Of course, we know — indeed this was a guiding principle throughout our
discussions — that the solution which we obtained by the preliminary
methods of 22., 23., are solutions in the sense of our present postulates
too. Nevertheless it is desirable to verify this explicitly. The real point,
however, is to ascertain whether the present postulates do not ascribe to
those games further solutions as well. (We have already seen that it is not
inconceivable that there should exist several solutions for the same game.)

We shall therefore determine all solutions for the essential zero-sum
three-person games — with results which are rather surprising, but, as we
shall see, not unreasonable.

1 If the relation S referred to at the end of 30.3.6. could be found, then this 3 — and
not (R — would disclose which standards of behavior are such nuclei (i.e. subsets): the
S-satisfactory ones.

Cf. the similar situation in 51.4., where the corresponding operation is performed
successfully.

272

GENERAL THEORY: ZERO-SUM n-PERSONS

30.4.2. These two items exhaust actually all zero-sum games with n £ 3.
We observed in the first remark of 27.5.2. that for n = 1,2, these games are
inessential; so this, together with the inessential and the essential cases of
n = 3, takes care of everything in n ^ 3.

When this program is fulfilled we are left with the games n ^ 4 — and
we know already that difficulties of a new kind begin with them (cf. the
allusions of footnote 1, p. 250, and the end of 27.5.3.).

30.4.3. Third: We introduced in 27.1. the concept of strategic equiva-
lence. It appeared plausible that this relationship acts as its name
expresses: two games which are linked by it offer the same strategical
possibilities and inducements to form coalitions, etc. Now that we have
put the concept of a solution on an exact basis, this heuristic expectation
demands a rigorous proof.

These three questions will be answered in (31 :P) in 31.2.3. ; in 32.2. ; and
in (31 :Q) in 31.3.3., respectively.

31. First Consequences

31.1. Convexity, Flatness, and Some Criteria for Domination

31.1.1. This section is devoted to proving various auxiliary results con-
cerning solutions, and the other concepts which surround them, like ines-
sentiality, essentiality, domination, effectivity. Since we have now put
all these notions on an exact basis, the possibility as well as the obligation
arises to be absolutely rigorous in establishing their properties. Some of
the deductions which follow may seem pedantic, and it may appear occasion-
ally that a verbal explanation could have replaced the mathematical proof.
Such an approach, however, would be possible for only part of the results
of this section and, taking everything into account, the best plan seems to
be to go about the whole matter systematically with full mathematical rigor.

Some principles which play a considerable part in finding solutions are
(31 :A), (31 :B), (31 :C), (31 :F), (31 :G), (31 :H), which for certain coalitions
decide a priori that they must always, or never, be taken into consideration.
It seemed appropriate to accompany these principles with verbal explana-
tions (in the sense indicated above) in addition to thqir formal proofs.

The other results possess varying interest of their own in different direc-
tions. Together they give a first orientation of the circumstances which
surround our newly won concepts. The answers to the first and third ques-
tions in 30.4. are given in (31 :P) and (31 :Q). Another question which
arose previously is settled in (31 :M).

31.1.2. Consider two imputations a , and assume that it has become

— ^ ^

necessary to decide whether a h p or not. This amounts to deciding
whether or not there exists a set S with the properties (30:4:a)-(30:4:c) in
30.1.1. One of these, (30:4:c) is

> fa for all i in S.

FIRST CONSEQUENCES

273

We call this the main condition. The two others, (30:4:a), (30:4:b), are
the 'preliminary conditions.

Now one of the major technical difficulties in handling this concept of
domination — i.e. in finding solutions V in the sense of 30.1.1. — is the presence
of these preliminary conditions. It is highly desirable to be able, so to say,
to short circuit them, i.e. to discover criteria under which they are certainly
satisfied, and others under which they are certainly not satisfied. In look-
ing for criteria of the latter type, it is by no means necessary that they should

— ►

involve non-fulfillment of the preliminary conditions for all imputations a —

it suffices if they involve it for all those imputations a which fulfill the main

condition for some other imputation . (Cf. the proofs of (31 :A) or (31 :F),
where exactly this is utilized.)

We are interested in criteria of this nature in connection with the ques-
tion of determining whether a given set V of imputations is a solution or not;
i.e. whether it fulfills the conditions (30:5:a), (30:5:b) — the condition

— ►

(30:5:c) — in 30.1.1. This amounts to determining which imputations 0
are dominated by elements of V.

Criteria which dispose of the preliminary conditions summarily, in the
situation described above, are most desirable if they contain no reference

at all to a , 1 * 2 i.e. if they refer to S alone. (Cf. (31 :F), (31 :G), (31 :H).) But

even criteria which involve a may be desirable. (Cf. (31 :A).) We shall

consider even a criterion which deals with S and a by referring to the
— ►

behavior of another a (Of course, both in V. Cf. (31 :B).)

In order to cover all these possibilities, we introduce the following
terminology:

We consider proofs which aim at the determination of all imputations

0 , which are dominated by elements of a given set of imputations V. We

are thus concerned with the relations a H fi (a in V), and the question
whether a certain set S meets our preliminary requirements for such a rela-
tion. We call S certainly necessary if we know (owing to the fulfillment by S

— ►

of some appropriate criterion) that S and a always meet the preliminary
conditions. We call a set S certainly unnecessary , if we know (again owing
to the fulfillment by S of some appropriate criterion, but which may now

involve other things too, cf. above) that the possibility that S and a meet

1 The point being that in our original definition of a 0 the preliminary conditions
refer to 5 and to « (but not to 0). Specifically: (30:4 :b) does.

* The hypothetical element of V, which should dominate 0 .

274

GENERAL THEORY: ZERO-SUM n-PERSONS

the preliminary conditions can be disregarded (because this never happens,
or for any other reason. Cf. also the qualifications made above).

These considerations may seem complicated, but they express a quite
natural technical standpoint. 1

We shall now give certain criteria of the certainly necessary and of the
certainly unnecessary characters. After each criterion we shall give a
verbal explanation of its content, which, it is hoped, will make our technique
clearer to the reader.

31 . 1 . 3 . First, three elementary criteria:

(31 :A) S is certainly unnecessary for a given a (in V) if there exists

an i in S with a, = v((i)).

Explanation: A coalition need never be considered if it does not promise
to every participant (individually) definitely more than he can get for
himself.

— >

Proof: If a fulfills the main condition for some imputation, then > ft.
— >

Since 0 is an imputation, so ft ^ v((i)). Hence a* > v((i)). This con-
tradicts on = v((t)).

(31 :B) S is certainly unnecessary for a given a (in V) if it is certainly

necessary (and being considered) for another a ' (in V), such that
(31:1) a'i ^ ai for all i in S.

Explanation: A coalition need not be considered if another one, which
has the same participants and promises every one (individually) at least
as much, is certain to receive consideration.

— ► — ►

Proof: Let a and 0 fulfill the main condition : ai > /?< for all i in S. Then
a ' and /S fulfill it also, by (31 :1), a' > ft for all i in S. Since /S and a' are

1 For the reader who is familiar with formal logic we observe the following:

The attributes c ‘certainly necessary” and “ certainly unnecessary” are of a logical
nature. They are characterized by our ability to show (by any means whatever) that a
certain logical omission will invalidate no proof (of a certain kind). Specifically: Let a

proof be concerned with the domination of a p by an element a of V. Assume that this

domination a H 0 occurring with the help of the set S ( a in V) be under consideration.

Then this proof remains correct if we treat S and a (when they possess the attribute in
question) as if they always (or never) fulfilled the preliminary conditions, — without our
actually investigating these conditions. In the mathematical proofs which we shall
carry out, this procedure will be applied frequently.

It can even happen that the same S will turn out (by the use of two different criteria)

—>

to be both certainly necessary and certainly unnecessary (for the same a , — e.g. for all of
them). This means merely that neither of the two omissions mentioned above spoils

— ►

any proof. This can happen, for instance, when a fulfills the main condition for no
imputation. (An example is obtained by combining (3 1 :F) and (3 1 :G) in the case described
in (31 :E:b). Another is pointed out in footnote 1 on p. 310, and in footnote 1 on p. 431.)

FIRST CONSEQUENCES

275

being considered, they thus establish that p is dominated by an element of
V, and it is unnecessary to consider S and a .

(31 :C) S is certainly unnecessary if another set T QS is certainly

necessary (and is being considered).

Explanation: A coalition need not be considered if a part of it is already
certain to receive consideration.

— ► — >

Proof : Let a (in V) and p fulfill the main condition for S ; then they will

obviously fulfill it a fortiori for Teg, Since T and a are being considered,

they thus establish that p is dominated by an element of V and it is unneces-
— >

sary to consider S and a .

31 . 1 . 4 . We now introduce some further criteria, and on a somewhat
broader basis than immediately needed. For this purpose we begin with
the following consideration:

For an arbitrary set S = (k h ' • ' , k p ) apply (25:5) in 25.4.1., with
Si — ( ky ), 4 • • , S p = (fc p ). Then

v(S) ^ v((fci)) + • • • + v((fc„))

obtains, i.e.

(31:2) v(S) £ X v((fc)).

k in S

The excess of the left-hand side of (31 :2) over the right hand side expresses
the total advantage (for all participants together) inherent in the formation
of the coalition S. We call this the convexity of S . If this advantage
vanishes, i.e. if

(31:3) v(S) = l v((fc)),

k in 3

then we call S flat.

The following observations are immediate:

(31 :D)
(31:D:a)
(31:D:b)
(31:D:c)
(31 :E)

(31:E:a)

(31:E:b)

(31:E:c)

The following sets are always flat:

The empty set,

Every one-element set,

Every subset of a flat set.

Any one of the following assertions is equivalent to the in-
essentiality of the game:

I = (1, • • • , n) is flat,

There exists an S such that both S and —S are flat,

Every S is flat.

Proof: Ad (31:D:a), (31:D:b): For these sets (31:3) is obvious.

276 GENERAL THEORY: ZERO-SUM n-PERSONS

Ad (31:D:c): Assume S sT, T flat. Put R — T — S. Then by (31:2)
(31:4) v(S) £ l v((k)),

kin 8

(31:5) v(ft) S: X ▼((*)).

k in R

Since T is flat, so by (30:3)

(31:6) v(T) = l v((fc)).

kin T

As S n R = ©, S u R = T ; therefore

v(S) + v(B) ^ v(r),

X v (( fc )) + X v (( fc )) = X

fc in S ^ in R k in T

Hence (31:6) implies

(31:7) v(,S) + v(R)g X v (( fc » + X v((*)).

A: in S A; in R

Now comparison of (31:4), (31:5) and (31:7) shows that we must have
equality in all of them. But equality in (31:4) expresses just the flatness
of 8.

Ad (31:E:a): The assertion coincides with (27 :B) in 27.4.1.

Ad (31:E:c): The assertion coincides with (27:C) :a 27.4.2.

Ad (31:E:b): For an inessential game this is true for any S owing to
(31 :E:c). Conversely, if this is true for (at least one) S, then

V(S) = X v (( fc ))> v(-S) = X v((*)) f

k in 8 k not in 8

hence by addition (use (25:3:b) in 25.3.1.),

0 = X ▼((*))»

h-l

i.e. the game is inessential by (31:E:a) or by (27 :B) in 27.4.1.

31 . 1 . 6 . We are now in a position to prove:

(31 :F) S is certainly unnecessary if it is flat.

Explanation: A coalition need not be considered if the game allows no
total advantage (for all its participants together) over what they would get
for themselves as independent players. 1

1 Observe that this is related to (3 1 : A) , but not at all identical with it ! Indeed : (31 : A)
deals with the o t , i.e. with the promises made to each participant individually. (31 :F)
deals with v(/S) (which determines flatness), i.e. with the possibilities of the game for all
participants together. But both criteria correlate these with the v((i)), i.e. with what
each player individually can get for himself.

FIRST CONSEQUENCES

277

Proof: If a (-> 0 with the help of this S then we have: Necessarily
S 9 * ©. off > pi for all i in S and ft ^ v((i)), hence a< > v((i)). So
2) a< > v((t)). As S is flat, this means £ <*< > v(S). But £ must

t in 8 % in 8 i in 5

be effective, ^ v(S), which is a contradiction.

tinS

(31 :G) S is certainly necessary if — S is flat and S Q.

Explanation: A coalition must be considered if it (is not empty and)
opposes one of the kind described in (31 :F).

Proof: The preliminary conditions are fulfilled for all imputations a .
Ad (30:4:a): S © was postulated.

Ad (30:4:b): Always a, ^ v((z)), so ^ ^ £ v((i)). Since

* not in 8 i not in S

n

5) = 0, the left-hand side is equal to — ^ cu. Since — S is flat, the

% — 1 * in S

right-hand side is equal to v( — S), i.e. (use (25:3:b) in 25.3.1.) to — v(£).
So — = — v(S), ^ ^ v(S), i.e. S is effective.

t in 5 t in S

From (31 :F), (31 :G) we obtain by specialization:

(31 :H) A p-element set is certainly necessary if p = n — 1, and

certainly unnecessary if p = 0, 1, n.

Explanation : A coalition must be considered if it has only one opponent.
A coalition need not be considered, if it is empty or consists of one player
only (!), or if it has no opponents.

Proof : p = n — 1: — S has only one element, hence it is flat by (31 :D)
above. The assertion now follows from (31 :G).

p = 0, 1: Immediate by (31 :D) and (31 :F).

p = n: In this case necessarily S = J = (l,***,n) rendering the
main condition unfulfillable. Indeed, that now requires a* > ft for all

n n

i = 1, • • • , n, hence £ on > ft.

t-i »»i

But as a , /3 are imputations, both

sides vanish, — and this is a contradiction.

Thus those p for which the necessity of S is in doubt, are restricted to
p 0, 1, n — 1, n, i.e. to the interval

(31:8) 2 g p g n - 2.

This interval plays a role only when n ^ 4. The situation discussed is
similar to that at the end of 27.5.2. and in 27.5.3., and the case n = 3
appears once more as one of particular simplicity.

31.2. The System of All Imputations. One-element Solutions
31 . 2 . 1 . We now discuss the structure of the system of all imputations.
(31 :1) For an inessential game there exists precisely one imputation:

278

GENERAL THEORY: ZERO-SUM n-PERSONS

(31:9) a = {ai, • • • •, a»), a* = v((i)) for i - 1, • • • , n.

For an essential game there exist infinitely many imputations
— an (n — l)-dimensional continuum — but (31 :9) is not one of
them.

Proof: Consider an imputation

P = {Pi, ■ '
Pi = v((i)) + a for

> $»!>

and put

* = 1, • • • , n.

Then the characteristic conditions (30:1), (30:2) of 30.1.1. become
(31:10) €< 0 for i = 1, • • • , n.

(31:11) X « = - X v((i)).

»-l »-l

If T is inessential, then (27 :B) in 27.4.1. gives — ^ v((i)) = 0; so

t-i

(31:10), (31:11) amount to *i = • • • = e n = 0, i.e. (31:9) is the unique
imputation.

n

If T is essential, the (27 :B) in 27.4.1. gives — v((t)) > 0, so (31:10),

»-i

(31 :11) possess infinitely many solutions, which form an (n — l)-dimensional

continuum; 1 so the same is true for the imputations 0 . But the a of (31 :9)
is not one of them, because ci = • • • = e n = 0 now violate (31:11).

An immediate consequence:

(31 :J) A solution V is never empty.

Proof: I.e. the empty set © is not a solution. Indeed: Consider any

>

imputation 0 , — there exists at least one by (31:1). 0 is not in © and no

a in © has a H 0 . So © violates (30:5:b) in 30.1.1. 2

31 . 2 . 2 . We have pointed out before that the simultaneous validity of

(31:12) a H p 9

is by no means impossible. 3 However:

0 h

(31 :K) Never a H a.

1 There is only one equation: (31:11).

‘This argument may seem pedantic; but if the conditions for the imputations con-
flicted (i.e. without (31:1)), then V — © would be a solution.

3 The sets S of these two dominations would have to be disjunct. By (31 :H) these
S must have ^ 2 elements each. Hence (31 :12) can occur only when n ^ 4.

By a more detailed consideration even n — 4 can be excluded; but for every n £ 5
(31:12) is really possible.

FIRST CONSEQUENCES

279

Proof: (30:4:a), (30:4:c) in 30.1.1. conflict for fa = /3 .

(31 :L) Given an essential game and an imputation a, there exists

— > — ► — ► — > — ►

an imputation fi such that 0 h a but not a H 0 . l

Proof: Put

a = {a,, • • • , a„}.

Consider the equation

(31:13) a< = v((0).

Since the game is essential, (31:1) excludes the proposition that (31:13) be

valid for all i = 1, • • • , n. Let (31:13) fail, say fori = i 0 . Since a is an
imputation, soa, o Si v((i 0 )), hence the failure of (31:13) means a,-, > v((i 0 )),
i.e.

(31:14) a< 0 = v((i 0 )) + «, e > 0.

Now define a vector

by

0 — {fill ’ ' ’ l 0n) >

Pi. = «<,-« = v((i 0 )),

Pi = ai H T for i t* i 0 .

n — 1

n n

These equations make it clear that ft ^ v((z)) 2 3 and that ^ ft = a» = 0. 8

So P is an imputation along with a .

— > — >

We now prove the two assertions concerning a , 0 .

—4 - — ►

P H a : We have & > a< for all i ^ i 0 , i.e. for all i in the set S = — ( io ) .
This set has n — 1 elements and it fulfills the main condition (for P , a ),

hence (31 :H) gives p & a .

Not a *-• 0 : Assume that a £•* P . Then a set S fulfilling the main con-
dition must exist, which is not excluded by (31 :H). So S must have
;> 2 elements. So an i ^ i 0 in S must exist. The former implies Pi > a»

1 Hence a 0 .

* For t — to, we have actually 0, o — v((t 0 )). For i ^ t 0 , we have 0* > a* £ v((i)).

n n

3 0< ^ «» because the difference of 0, and a,- is c for one value of i (t — i 0 ) and

»-i »-i

for w — 1 values of t (all i ^ to).

n — 1

280

GENERAL THEORY: ZERO-SUM n-PERSONS

(by the construction of p ) ; the latter implies a» > ft (owing to the main
condition) — and this is a contradiction.

31 . 2 . 3 * We can draw the conclusions in which we were primarily inter-
ested:

(31 :M) An imputation a , for which never a f H a, exists if and

only if the game is inessential. 1

Proof: Sufficiency: If the game is inessential, then it possesses by (31:1)
precisely one imputation a , and this has the desired property by (31 :K).

Necessity: If the game is essential, and a is an imputation, then a ' =
of (31 :L) gives a' = 0 H a.

(31 :N) A game which possesses a one-element solution 2 is necessarily

inessential.

— ►

Proof: Denote the one-element solution in question by V = ( a ). This
V must satisfy (30:5:b) in 30.1.1. This means under our present circum-
— > — > — ►

stances: Every other than a is dominated by a. I.e. :

P a implies a H j8,

Now if the game is essential, then (31 :L) provides a which violates this
condition.

(31 :0) An inessential game possesses precisely one solution V. This

— ► — >

is the one-element set V = ( a ) with the a of (31:1).

— >

Proof: By (31:1) there exists precisely one imputation, the a of (31:1).
A solution V cannot be empty by (31 :J); hence the only possibility is

V = («). NowV = ( a ) is indeed a solution, i.e. it fulfills (30:5 :a), (30:5:b)

in 30.1.1. : the former by (31 :K), the latter because a is the only imputation
by (31:1).

We can now answer completely the first question of 30.4.1.:

(31 :P) A game possesses a one-element solution (cf. footnote 2 above)

if and only if it is inessential; and then it possesses no other
solutions.

Proof: This is just a combination of the results of (31 :N) and (31:0).

1 Cf. the considerations of (30:A:a) in 30.2.2., and particularly footnote 2 on p. 265.
* We do not exclude the possibility that this game may possess other solutions as well,
which may or may not be one-element sets. Actually this never happens (under our
present hypothesis), as the combination of the result of (31 :N) with that of (31:0) — or
the result of (31 :P)— shows. But the present consideration is independent of all this.

FIRST CONSEQUENCES

281

31.3. The Isomorphism Which Corresponds to Strategic Equivalence

31.3.1. Consider two games T and r' with the characteristic functions
v(S) and v'(S) which are strategically equivalent in the sense of 27.1.
We propose to prove that they are really equivalent from the point of view
of the concepts defined in 30.1.1. This will be done by establishing an
isomorphic correspondence between the entities which form the substratum
of the definitions of 30.1.1., i.e. the imputations. That is, we wish to
establish a one-to-one correspondence, between the imputations of T and
those of T', which is isomorphic with respect to those concepts, i.e. which
carries effective sets, domination, and solutions for T into those for T'.

The considerations are merely an exact elaboration of the heuristic
indications of 27.1.1., hence the reader may find them unnecessary. How-
ever, they give quite an instructive instance of an “ isomorphism proof,”
and, besides, our previous remarks on the relationship of verbal and of
exact proofs may be applied again.

31.3.2. Let the strategic equivalence be given by a®, • • , a® m the sense

of (27:1), (27:2) in 27.1.1. Consider all imputations a = {ai, * * • , a n |

of T and all imputations a ' — \a ■ • • , a^ofT'. We look for a one-to-
one correspondence

(31:15)

r

with the specified properties.

What (31:15) ought to be is easily guessed from the motivation at the
beginning of 27.1.1. We described there the passage from T to T f by adding
to the game a fixed payment of a® to the player k. Applying this principle
to the imputations means

(31:16) a' k = ot k + a® for k = 1, • * • , n. 1

Accordingly we define the correspondence (31:15) by the equations (31:16).
31.3.3. We now verify the asserted properties of (31:15), (31:16).

The imputations of V are mapped on the imputations of T': This means
by (30:1), (30:2) in 30.1.1., that

(31:17)

(31:18)

go over into
(31:17*)

(31:18*)

a t ^ v((i)) for i = 1, • • • . n.

X a ‘ = °i

»-i

a[ ^ v'((i)) for i = 1, • • • , n,

i < = o.

t«i

1 If we introduce the (fixed) vector a 0 ■* | af, ♦ • ♦ , aj) then (31:16) may be

written vectorially a ' * a -f a °. I.e. it is a translation (by a ) in the vector space
of the imputations.

282 GENERAL THEORY: ZERO-SUM n-PERSONS

This is so for (31:17), (31:17*) because v'((i)) = v((t)) + a? (by (27:2) in

n

27.1.1. ), and for (31:18), (31:18*) because ^ a? = 0 (by (27:1) id.).

i-1

Effectivity for T goes over into effectivity for F' : This means by (30:3)
in 30.1.1., that

X «. ^ v(<s)

i in 8

goes over into

X ^ vw

» in S

This becomes evident by comparison of (31:16) with (27:2).

Domination for r goes over into domination for T' : This means the same
thing for (30:4:a)-(30:4:c) in 30.1.1. (30:4:a) is trivial; (30:4:b) is effec-

tivity, which we settled: (30:4:c) asserts that a* > ft goes over into a\ >
which is obvious. The solutions of r are mapped on the solutions of r':
This means the same for (30:5:a), (30:5:b) (or (30:5:c)) in 30.1.1. These
conditions involve only domination, which we settled.

We restate these results:

(31 :Q) If two zero-sum games r and T' are strategically equivalent,

then there exists an isomorphism between their imputations —
i.e. a one-to-one mapping of those of T on those of r' which
leaves the concepts defined in 30.1.1. invariant.

32. Determination of all Solutions
of the Essential Zero-sum Three-person Game

32.1. Formulation of the Mathematical Problem. The Graphical Method

32.1.1. We now turn to the second problem formulated in 30.4.1.:
The determination of all solutions for the essential zero-sum three-person
games.

We know that we may consider this game in the reduced form and that
we can choose 7 = l. 1 The characteristic function in this case is completely
determined as we have discussed before: 2

/ 0

(32:1) v(S) = 1 | when S has

( 0

An imputation is a vector

— >

a = {«i, at, «»}

1 Cf. the discussion at the beginning of 29.1., or the references there given: the end of

27.1. and the second remark in 27.3.

* Cf. the discussion at the beginning of 29.1., or the second case of 27.5.

0

2 elements.

3

ALL SOLUTIONS OF THE THREE-PERSON GAME 283

whose three components must fulfill (30:1), (30:2) in 30. 1.1.* These con-
ditions now become (considering (30:1))

(32:2) on — 1, a 2 ^ — 1, as — 1,

(32:3) «i + an + as = 0.

We know, from (31:1) in 31.2.1., that these a h a 2 , a 8 form only a two-
dimensional continuum — i.e. that they should be representable in the plane.
Indeed, (32:3) makes a very simple plane representation possible.

32.1.2. For this purpose we take three axes in the plane, making angles
of 60° with each other. For any point of the plane we define an, an, an
by directed perpendicular distances from these three axes. The whole
arrangement, and in particular the signs to be ascribed to the an, an, ots

284

GENERAL THEORY: ZERO-SUM n-PERSONS

are given in Figure 52. It is easy to verify that for any point the algebraic
sum of these three perpendicular distances vanishes and that conversely

any triplet a = {«i, <* 2 , ots] for which the sum vanishes, corresponds to a
point.

So the plane representation of Figure 52 expresses precisely the condition
(32:3). The remaining condition (32:2) is therefore the equivalent of a

restriction imposed upon the point a within the plane of Figure 52. This
restriction is obviously that it must lie on or within the triangle formed
by the three lines <*i= — 1, a 2 = — 1, = — 1. Figure 53 illustrates this.

Thus the shaded area, to be called the fundamental triangle , represents
• >

the a which fulfill (32:2), (32:3) — i.e. all imputations.

32 . 1 . 3 . We next express the relationship of domination in this graphical
representation. As n = 3, we know from (31 :H) (cf. also the discussion of

(31:8) at the end of 31.1.5.) that among the subsets S of / = (1, 2, 3)
those of two elements are certainly necessary, and all others certainly
unnecessary. I.e., the sets which we must consider in our determination
of all solutions V are precisely these :

(1,2); (1,3); (2,3).

Thus for

— > — >

a = {ai, at, at}, fi = {fii, fit, fit]

domination

~a H ~fi

means that

(32:4) Either ai > j8i, 02 > fit', or ai > fii, a 8 > fit', or aa > fit,

at > fit-

Diagrammatically: a dominates the points in the shaded areas, and no
others, 1 in Figure 54.

1 In particular, no points on the boundary lines of these areas.

ALL SOLUTIONS OF THE THREE-PERSON GAME 285

Thus the point a dominates three of the six sextants indicated in

Figure 55 (namely A, C, E). From this one concludes easily that a is
dominated by the three other sextants (namely B 1 D , F). So the only

points which do not dominate a and are not dominated by it, lie on the
three lines (i.e. six half-lines) which separate these sextants. I.e. :

(32:5) If neither of a , 0 dominates the other, then the direction

— > — *

from a to 0 is parallel to one of the sides of the fundamental
triangle.

32 . 1 . 4 . Now the systematic search for all solutions can begin.

Consider a solution V, i.e. a set in the fundamental triangle which
fulfills the conditions (30:5:a), (30:5:b) of 30.1.1. In what follows we shall
use these conditions currently, without referring to them explicitly on each
occasion.

Since the game is essential, V must contain at least two points 1 say

a and /S . By (32:5) the direction from a to is parallel to one of the
sides of the fundamental triangle; and by a permutation of the numbers of the
players 1,2,3 we can arrange this to be the side = — 1, i.e. the horizontal.

So a , £ lie on a horizontal line l. Now two possibilities arise and we treat
them separately:

(a) Every point of V lies on l.

(b) Some points of V do not lie on L

32.2. Determination of Ail Solutions

32 . 2 . 1 . We consider (b) first. Any point not on l must fulfill (32:5) with
— ► — ►

respect to both a and 0 , i.e. it must be the third vertex of one of the two

equilateral triangles with the base a , @ : one of the two points a ', a " of

Figure 56. So either a ' or a " belongs to V. Any point of V which differs
— ► — > — > — ►

from a, 0 and a 'or a " must again fulfill (32:5), but now with respect to all

three points a , $ and a ' or a ". This, however, is impossible, as an
inspection of Figure 56 immediately reveals. So V consists of precisely these
three points, — i.e. of the three vertices of a triangle which is in the position
of triangle I or triangle II of Figure 57. Comparison of Figure 57 with
Figures 54 or 55 shows that the vertices of triangle I leave the interior of
this triangle uridominated. This rules out I. 2

1 This is also directly observable in Fig. 54.

2 This provides the example referred to in 30.3.6.: The three vertices of triangle I do
not dominate each other, i.e. they form a satisfactory set in the sense of loc. cit. They are
nevertheless unsuitable as a subset of a solution.

ALL SOLUTIONS OF THE THREE-PERSON GAME 287

The same comparison shows that the vertices of triangle II leave
undominated the dotted areas indicated in Figure 58. Hence triangle II
must be placed in such a manner in the fundamental triangle that these
dotted areas fall entirely outside the fundamental triangle. This means
that the three vertices of II must lie on the three sides of the fundamental
triangle, as indicated in Figure 59. Thus these three vertices are the
middle points of the three sides of the fundamental triangle.

Comparison of Figure 59 with Figure 54 or Figure 55 shows that this
set V is indeed a solution. One verifies immediately that these three
middle points are the points (vectors)

(32:6) {-!,*, ii, {*, -1,*}, {*,*, -1),

i.e. that this solution V is the set of Figure 51.

32.2.2. Let us now consider (a) in 32.1.4. In this case all of V lies on the
horizontal line L By (32 :5) no two points of l dominate each other, so that
every point of l is undominated by V. Hence every point of l (in the funda-
mental triangle) must belong to V. I.e., V is precisely that part of l

which is in the fundamental triangle. So the elements a = [a Xy a 2 , a 8 }
of V are characterized by an equation

(32:7) ai = c.

Diagrammatically: Figure 60.

Comparison of Figure 60 with Figures 54 or 55 shows that the line l
leaves the dotted area indicated on Figure 60 undominated. Hence the line
l must be placed in such a manner in the fundamental triangle that the
dotted area falls entirely outside the fundamental triangle. This means
that l must lie below the middle points of those two sides of the fundamental

288

GENERAL THEORY: ZERO-SUM n-PERSONS

triangle which it intersects. 1 In the terminology of (32:7): c < |. On the
other hand, c ^ — I is necessary to make l intersect the fundamental
triangle at all. So we have:

(32:8) -Uc<i

Comparison of Figure 60 with Figures 54 or 55 shows that under these
conditions 2 the set V — i.e. I — is indeed a solution.

But the form (32:7) of this solution was brought about by a suitable
permutation of the numbers 1,2,3. Hence we have two further solutions,
characterized by

(32:7*) a 2 = c,

and characterized by

(32:7**) a 3 = c,

always with (32:8)

32.2.3. Summing up :

This is a complete list of solutions :

(32:A) For every c which fulfills (32:8): The three sets (32:7)

(32:7*), (32:7**).

(32 :B) The set (32:6).

33. Conclusions

33.1. The Multiplicity of Solutions. Discrimination and Its Meaning

33.1.1. The result of 32. calls for careful consideration and comment.
We have determined all solutions of the essential zero-sum three-person
game. In 29.1., before the rigorous definitions of 30.1. were formulated,
we had already decided which solution we wanted; and this solution reap-
pears now as (32 :B). But we found other solutions besides: the (32 :A),
which are infinitely many sets, each one of them an infinite set of imputa-
tions itself. What do these supernumerary solutions stand for?

Consider, e.g., the form (32:7) of (32 :A). This gives a solution for

every c of (32:8) consisting of all imputations a = [a h a 2 , « 3 } which fulfill
(32:7), i.e. ai = c. Besides this, they must fulfill only the requirements,

1 The limiting position of l, going through the middle points themselves, must be
excluded. The reason is that in this position the vertex of the dotted area would lie on
the fundamental triangle, — and this is inadmissable since that point too is undominated
by V, i.e. by l.

Observe that the corresponding prohibition did not occur in case (b), i.e. for the
dotted areas of Figure 58. Their vertices too were undominated by V, but they belong
to V. In our present position of V, on the other hand, the vertex under consideration
does not belong to V, i.e. to l.

This exclusion of the limiting position causes the < — and not the £ — in the
inequality which follows.

* (32:8), i.e. I intersects the fundamental triangle, but below its middle.

CONCLUSIONS 289

(30:1), (30:2) of 30.1.1. i.e. (32:2), (32:3) of 32.1.1. In other words:
Our solution consists of all

(33:1) a = {c, a, — c —a}, — l^a^l — c.

The interpretation of this solution consists manifestly of this: One
of the players (in this case 1) is being discriminated against by the two others
(in this case 2,3). They assign to him the amount which he gets, c. This
amount is the same for all imputations of the solution, i.e. of the accepted
standard of behavior. The place in society of player 1 is prescribed by the
two other players; he is excluded from all negotiations that may lead to
coalitions. Such negotiations do go on, however, between the two other
players: the distribution of their share, — c, depends entirely upon their
bargaining abilities. The solution, i.e. the accepted standard of behavior,
imposes absolutely no restrictions upon the way in which this share is
divided between them, — expressed by a, —c — a. 1 2 * 4 This is not surprising.
Since the excluded player is absolutely “tabu,” the threat of the partner's
desertion is removed from each participant of the coalition. There is no
way of determining any definite division of the spoils. 2,3

Incidentally: It is quite instructive to see how our concept of a solution
as a set of imputations is able to take care of this situation also.

33.1.2. There is more that should be said about this “discrimination”
against a player.

First, it is not done in an entirely arbitrary manner. The c, in which
discrimination finds its quantitative expression, is restricted by (32:8) in
32.2.2. Now part of (32:8), c ^ — 1, is clear enough in its meaning, but the
significance of the other part c < i 4 is considerably more recondite (cf.
however, below). It all comes back to this: Even an arbitrary system of
discriminations can be compatible with a stable standard of behavior — i.e.
order of society — but it may have to fulfill certain quantitative conditions,
in order that it may not impair that stability.

Second, the discrimination need not be clearly disadvantageous to the
player who is affected. It cannot be clearly advantageous, — i.e. his fixed
value c cannot be equal to or better than the best the others may expect.
This would mean, by (33:1), that c ^ 1 — c, i.e. c ^ *, — which is exactly
what (32:8) forbids. But it would be clearly disadvantageous only for
c = — 1 ; and this is a possible value for c (by (32 :8)), but not the only one.
c = — 1 means that the player is not only excluded, but also exploited to

1 Except that both must be — 1 — i.e. what the player can get for himself, without

any outside help.

a, — c — a —1 is, of course, the —1 £ a £ 1 — cof (33:1).

2 Cf. the discussions at the end of 25.2. Observe that the arguments which we
adduced there to motivate the primate of v(S) have ceased to operate in this particular
case — and v(<S) nevertheless determines the solutions!

8 Observe that due to (32:8) in 32.2.2., the “spoils”, i.e. the amount — c, can be both
positive and negative.

4 And that is excluded in c < J, but not in c ^ —1.

290

GENERAL THEORY: ZERO-SUM n-PERSONS

100 per cent. The remaining c (of (32:8)) with —1 < c < i correspond to
gradually less and less disadvantageous forms of segregation.

33 . 1 . 3 . It seems remarkable that our concept of a solution is able to
express all these nuances of non-discriminatory (32 :B), and discriminatory
(32 :A), standards of behavior — the latter both in their 100 per cent injurious
form, c = — 1, and in a continuous family of less and less injurious ones
— 1 < c < i. It is particularly significant that we did not look for any
such thing — the heuristic discussions of 29.1 were certainly not in this spirit
— but we were nevertheless forced to these conclusions by the rigorous theory
itself. And these situations arose even in the extremely simple framework
of the zero-sum three-person game!

For n^4 we must expect a much greater wealth of possibilities for
all sorts of schemes of discrimination, prejudices, privileges, etc. Besides
these, we must always look out for the analogues of the solution (32:B), i.e.
the nondiscriminating “ objective ” solutions. But we shall see that the con-
ditions are far from simple. And we shall also see that it is precisely the
investigation of the discriminatory “inobjective” solutions which leads to a
proper understanding of the general non-zero-sum games — and thence to
application to economics.

33.2. Statics and Dynamics

33 . 2 . At this point it may be advantageous to recall the discussions of
4.8.2. concerning statics and dynamics. What we said then applies now;
indeed it was really meant for the phase which our theory has now reached.

In 29.2. and in the places referred to there, we considered the nego-
tiations, expectations and fears which precede the formation of a coalition
and which determine its conditions. These were all of the quasi-dynamic
typo described in 4.8.2. The same applies to our discussion in 4.6. and
again in 30.2., of how various imputations may or may not dominate each
other depending on their relationship to a solution; i.e., how the conducts
approved by an established standard of behavior do not conflict with each
other, but can be used to discredit the non-approved varieties.

The excuse, and the necessity, for using such considerations in a static
theory were set forth on that occasion. Thus it is not necessary to repeat
them now.

CHAPTER VII

ZERO-SUM FOUR-PERSON GAMES

34. Preliminary Survey
34.1. General Viewpoints

34.1. We are now in possession of a general theory of the zero-sum
n-person game, but the state of our information is still far from satisfactory.
Save for the formal statement of the definitions we have penetrated but
little below the surface. The applications which we have made — i.e. the
special cases in which we have succeeded in determining our solutions — can
be rated only as providing a preliminary orientation. As pointed out in
30.4.2., these applications cover all cases n ^ 3, but we know from our past
discussions how little this is in comparison with the general problem. Thus
we must turn to games with n ^ 4 and it is only here that the full complexity
of the interplay of coalitions can be expected to emerge. A deeper insight
into the nature of our problems will be achieved only when we have mastered
the mechanisms which govern these phenomena.

The present chapter is devoted to zero-sum four-person games. Our
information about these still presents many lacunae. This compels an
inexhaustive and chiefly casuistic treatment, with its obvious shortcomings. 1
But even this imperfect exposition will disclose various essential qualitative
properties of the general theory which could not be encountered previously,
(for n ^ 3). Indeed, it will be noted that the interpretation of the mathe-
matical results of this phase leads quite naturally to specific “social”
concepts and formulations.

34.2. Formalism of the Essential Zero-sum Four-person Game

34.2.1. In order to acquire an idea of the nature of the zero-sum four-
person games we begin with a purely descriptive classification.

Let therefore an arbitrary zero-sum four-person game T be given, which
we may as well consider in its reduced form: and also let us choose y = l. 2
These assertions correspond, as we know from (27:7*) and (27:7**) in 27.2.,
to the following statements concerning the characteristic functions:

/ ° / 0

(34:1) , v(S) = < J when S has < * elements.

( 0 ( 4

1 E.g., a considerable emphasis on heuristic devices.

2 Cf. 27.1.4. and 27.3.2. The reader will note the analogy between this discussion
and that of 29.1.2. concerning the zero-sum three-person game. About this more will
be said later.

291

292

ZERO-SUM FOUR-PERSON GAMES

Thus only the v(S) of the two-element sets S remain undetermined by
these normalizations. We therefore direct our attention to these sets.

The set I = (1,2, 3, 4) of all players possesses six two-element subsets S :

(1,2), (1,3), (1,4), (2,3), (2,4), (3,4).

Now the v(S ) of these sets cannot be treated as independent variables,
because each one of these S has another one of the same sequence as its
complement. Specifically: the first and the last, the second and the fifth,
the third and the fourth, are complements of each other respectively.
Hence their v(S) are the negatives of each other. It is also to be remem-
bered that by the inequality (27:7) in 27.2. (with n — 4, p = 2) all these
y(S) are £2, ^ —2. Hence if we put

v((l,4)) = 2xi,

(34:2) v((2,4)) = 2x 2 ,

v((3,4)) = 2 x 3 ,

then we have

v((2,3)) = -2xi,

(34:3) v((l,3)) = -2x 2 ,

v((l,2)) = ~2x 3 ,

and in addition

(34:4) —1 ^ xi, x 2 , x 3 ^ 1.

Conversely: If any three numbers Xi, x 2 , x 3 fulfilling (34:4) are given,
then we can define a function y(S) (for all subsets S of I = (1,2, 3, 4)) by
(34:l)-(34:3), but we must show that this v(*S) is the characteristic function
of a game. By 26.1. this means that our present y(S) fulfills the conditions
(25:3:a)-(25:3:c) of 25.3.1. Of these, (25:3:a) and (25:3:b) are obviously
fulfilled, so only (25:3:c) remains. By 25.4.2. this means showing that

v(Si) + y(S 2 ) + y(Sz) ^ 0 if Si, S 2 , S 8 are a decomposition of I.

(Cf. also (25:6) in 25.4.1.) If any of the sets Si, S 2 , S 8 is empty, the two
others are complements and so we even have equality by (25:3:a), (25:3:b)
in 25.3.1. So we may assume that none of the sets Si, S 2 , S 8 is empty.
Since four elements are available altogether, one of these sets, say S\ — S,
must have two elements, while the two others are one-element sets. Thus
our inequality becomes

y(S) -2^0, i.e. v(fl) ^ 2.

If we express this for all two-element sets S, then (34:2), (34:3) transform
the inequality into

2xi S 2, 2x2 ^ 2, 2xs ^ 2,

— 2xi 2, — 2 x 2 ^ 2, 2x 3 ^ 2,

which is equivalent to the assumed (34 :4) . Thus we have demonstrated :

PRELIMINARY SURVEY

293

(34: A) The essential zero-sum four-person games (in their reduced

form with the choice 7 = 1) correspond exactly to the triplets
of numbers x h x 2y £ 3 fulfilling the inequalities (34:4). The
correspondence between such a game, i.e. its characteristic
function, and its X\> x 2 , x 3 is given by the equations (34:l)-(34:3). 1

34.2.2. The above representation of the essential zero-sum four-person
games by triplets of numbers x h x 2) x% can be illustrated by a simple geo-
metrical picture. We can view the numbers x h x 2) x 3 as the Cartesian
coordinates of a point. 2 In this case the inequalities (34:4) describe a part

+*»

Figure 61.

of space which exactly fills a cube Q. This cube is centered at the origin
of the coordinates, and its edges are of length 2 because its six faces are the
six planes

£1 = ±1, £2 = ±1, £3 — ±1,

as shown in Figure 61.

Thus each essential zero-sum four-person game T is represented by
precisely one point in the interior or on the surface of this cube, and vice
versa. It is quite useful to view these games in this manner and to try to
correlate their peculiarities with the geometrical conditions in Q. It
will be particularly instructive to identify those games which correspond to
definite significant points of Q.

1 The reader may now compare our result with that of 29.1.2. concerning the zero-sum
three-person games. It will be noted how the variety of possibilities has increased.

2 We may also consider these numbers as the components of a vector in Lj in the
sense of 16.1.2. et seq. This aspect will sometimes be the more convenient, as in foot-
note 1 on p. 304.

294

ZERO-SUM FOUR-PERSON GAMES

But even before carrying out this program, we propose to consider
certain questions of symmetry. We want to uncover the connections
between the permutations of the players 1,2, 3, 4, and the geometrical trans-
formations (motions) of Q. Indeed: by 28.1. the former correspond to the
symmetries of the game T, while the latter obviously express the symmetries
of the geometrical object.

34.3. Permutations of the Players

34 . 3 . 1 . In evolving the geometrical representation of the essential
zero-sum four-person game we had to perform an arbitrary operation,
i.e. one which destroyed part of the symmetry of the original situation.
Indeed, in describing the v(S) of the two-element sets £, we had to single
out three from among these sets (which are six in number), in order to
introduce the coordinates x X} x 2} x 8 . We actually did this in (34:2), (34:3)
by assigning the player 4 a particular role and then setting up a correspond-
ence between the players 1,2,3 and the quantities xi, x 2 , x 8 respectively
(cf. (34:2)). Thus a permutation of the players 1,2,3 will induce the same
permutation of the coordinates x Xy x 2 , x 8 — and so far the arrangement is
symmetric. But these are only six permutations from among the total of
24 permutations of the players 1,2, 3,4^ So a permutation which replaces
the player 4 by another one is not accounted for in this way.

34 . 3 . 2 . Let us consider such a permutation. For reasons which will
appear immediately, consider the permutation A, which interchanges the
players 1 and 4 with each other and also the players 2 and 3. 2 A look at the
equations (34:2), (34:3) suffices to show that this permutation leaves x x
invariant, while it replaces x 2 , x 8 by — x 2 , — x 3 . Similarly one verifies:
The permutation B , which interchanges 2 and 4, and also 1 and 3, leaves x 2
invariant and replaces x X) x 8 by — x X) — x 3 . The permutation C, which
interchanges 3 and 4 and also 1 and 2, leaves x 3 invariant and replaces Xi, x 2
by X\ y x 2 .

Thus each one of the three permutations A, B, C affects the variables
Xit x 2 , xz only as far as their signs are concerned, each changing two signs
and leaving the third invariant.

Since they also carry 4 into 1,2,3, respectively, they produce all permuta-
tions of the players 1,2, 3, 4, if combined with the six permutations of the
players 1,2,3. Now we have seen that the latter correspond to the six
permutations of x Xf x 2 , x 8 (without changes in sign). Consequently the
24 permutations of 1,2, 3, 4 correspond to the six permutations of x Xf x 2 , x 8 ,
each one in conjunction with no change of sign or with two changes of
sign. 3

1 Cf. 28.1.1., following the definitions (28:A:a), (28:A:b).

* With the notations of 29.1.:

3 These sign changes are 1+3-4 possibilities in each case, so we have 6 X 4 * 24
operations on x X} x 8y to represent the 24 permutations of 1,2, 3, 4, — as it should be.

SPECIAL POINTS IN THE CUBE Q

295

34.3.3. We may also state this as follows: If we consider all movements
in space which carry the cube Q into itself, it is easily verified that they
consist of the permutations of the coordinate axes xi, X 2 , x 8 in combination
with any reflections on the coordinate planes (i.e. the planes x*, x 3 ; xi, x*;
x \ , x%). Mathematically these are the permutations of xi, xj, x% in combina-
tion with any changes of sign among the x if x*, x*. These are 48 possibili-
ties. 1 Only half of these, the 24 for which the number of sign changes is
even (i.e. 0 or 2), correspond to the permutations of the players.

Figure 62 .

It is easily verified that these are precisely the 24 which not only carry
the cube Q into itself, but also the tetrahedron I, V, VI, VII, as indicated in
Figure 62. One may also characterize such a movement by observing that
it always carries a vertex • of Q into a vertex •; and equally a vertex o into a
vertex o, but never a • into a o . 2

We shall now obtain a much more immediate interpretation of these
statements by describing directly the games which correspond to specific
points of the cube Q : to the vertices • or o, to the center (the origin in Figure
61), and to the main diagonals of Q.

36. Discussion of Some Special Points in the Cube Q
36.1. The Corner / (and V , VI, VII)

36.1.1. We begin by determining the games which correspond to the
four corners •: I, V , VI, VII. We have seen that they arise from each other
by suitable permutations of the players 1,2, 3, 4. Therefore it suffices to
consider one of them, say /.

1 For each variable x\, x%, x% there are two possibilities: change or no change. This
gives altogether 2* — 8 possibilities. Combination with the six permutations of Xi, x%, x%
yields 8 X 6 — 48 operations.

a This group of motions is well known in group theory and particularly in crystallog-
raphy, but we shall not elaborate the point further.

296

ZERO-SUM FOUR-PERSON GAMES

The point 1 corresponds to the values 1,1,1 of the coordinates x lf x 2 , x 3 .
Thus the characteristic function v(S) of this game is:

1 0 / 0

-i 1 1

2 j 2 (and 4 belongs to S)

when S has \ elements

— 2 1 2 (and 4 does not belong to S)

1 ( 3

0 \ 4

(Verification is immediate with the help of (34:1), (34:2), (34:3) in 34.2.1.)
Instead of applying the mathematical theory of Chapter VI to this game, let
us first see whether it does not allow an immediate intuitive interpretation.

Observe first that a player who is left to himself loses the amount —1.
This is manifestly the worst thing that can ever happen to him since he
can protect himself against further losses without anybody else's help. 1
Thus we may consider a player who gets this amount — 1 as completely
defeated. A coalition of two players may be considered as defeated if it
gets the amount —2, since then each player in it must necessarily get — l. 2,3
In this game the coalition of any two players is defeated in this sense if it
does not comprise player 4.

Let us now pass to the complementary sets. If a coalition is defeated
in the above sense, it is reasonable to consider the complementary set
as a winning coalition. Therefore the two-element sets which contain the
player 4 must be rated as winning coalitions. Also since any player who
remains isolated must be rated as defeated, three-person coalitions always
win. This is immaterial for those three-element coalitions which contain
the player 4, since in these coalitions two members are winning already
if the player 4 is among them. But it is essential that 1,2,3 be a winning
coalition, since all its proper subsets are defeated. 4

1 This view of the matter is corroborated by our results concerning the three-person
game in 23. and 32.2., and more fundamentally by our definition of the imputation in
30.1.1., particularly condition (30:1).

2 Since neither he nor his partner need accept less than — 1, and they have together
— 2, this is the only way in which they can split.

8 In the terminology of 31.1.4.: this coalition is flat. There is of course no gain, and
therefore no possible motive for two players to form such a coalition. But if it happens
that the two other players have combined and show no desire to acquire a third ally, we
may treat the remaining two as a coalition even in this case.

4 We warn the reader that, although we have used the words “defeated” and “ win-
ning” almost as termini technici f this is not our intention. These concepts are, indeed,
very well suited for an exact treatment. The “defeated” and “ winning ” coalitions
actually coincide with the sets S considered in (31 :F) and in (31 :G) in 31.1.5.; those for
which S is flat or — S is flat, respectively. But we shall consider this question in such
a way only in Chap. X.

For the moment our considerations are absolutely heuristic and ought to be taken
in the same spirit as the heuristic discussions of the zero-sum three-person game in 21., 22.
The only difference is that we shall be considerably briefer now, since our experience and
routine have grown substantially in the discussion.

As we now possess an exact theory of solutions for games already, we are under

SPECIAL POINTS IN THE CUBE Q 297

35 . 1 . 2 . So it is plausible to view this as a struggle for participation in
any one of the various possible coalitions:

(35:2) (1,4), (2,4), (3,4), (1,2,3),

where the amounts obtainable for these coalitions are:

(35:3) v((l,4)) = v((2,4)) = v((3,4)) = 2, v((l,2,3)) = 1.

Observe that this is very similar to the situation which we found in
the essential zero-sum three-person game, where the winning coalitions
were:

(35:2*) (1,2), (1,3), (2,3),

and the amounts obtainable for these coalitions:

(35:3*) v((l,2)) = v((l,3)) = v((2,3)) = 1.

In the three-person game we determined the distribution of the pro-
ceeds (35:3*) among the winners by assuming: A player in a winning coali-
tion should get the same amount no matter which is the winning coalition.
Denoting these amounts for the players 1,2,3 by a, 0, y respectively, (35:3*)
gives

(35:4*) a + /? = a + Y = /3 + 7 = l

from which follows

(35:5*) « = 0 = y = i

These were indeed the values which those considerations yielded.

Let us assume the same principle in our present four-person game.
Denote by a, /?, 7 , 5, respectively, the amount that each player 1,2, 3, 4 gets
if he succeeds in participating in a winning coalition. Then (35:3) gives

(35:4) a + 6 = 0 + 5 = 7 + S = 2, a + 0 + 7 = 1,

from which follows

(35:5) a = f3 = 7 = i 5 = f.

All the heuristic arguments used in 21., 22., for the three-person game could
be repeated. 1

35 . 1 . 3 . Summing up:

(35 :A) This is a game in which the player 4 is in a specially favored

position to win: any one ally suffices for him to form a winning
coalition. Without his cooperation, on the other hand, three
players must combine. This advantage also expresses itself in

obligation to follow up this preliminary heuristic analysis by an exact analysis which
is based rigorously on the mathematical theory. We shall come to this. (Cf. loc. cit.
above, and also the beginning of 36.2.3.)

1 Of course, without making this thereby a rigorous discussion on the basis of 30.1.

298

ZERO-SUM FOUR-PERSON GAMES

the amounts which each player 1,2, 3, 4 should get when he is
among the winners — if our above heuristic deduction can be
trusted. These amounts are £, i, £, 1 respectively. It is
to be noted that the advantage of player 4 refers to the case of
victory only; when defeated, all players are in the same position
(i.e. get -1).

The last mentioned circumstance is, of course, due to our normaliza-
tion by reduction. Independently of any normalization, however, this game
exhibits the following trait: One player’s quantitative advantage over
another may, when both win, differ from what it is when both lose.

This cannot happen in a three-person game, as is apparent from the
formulation which concludes 22.3.4. Thus we get a first indication of an
important new factor that emerges when the number of participants reaches
four.

35 . 1 . 4 . One last remark seems appropriate. In this game player 4’s
strategic advantage consisted in the fact that he needed only one ally
for victory, whereas without him a total of three partners was necessary.
One might try to pass to an even more extreme form by constructing a game
in which every coalition that does not contain player 4 is defeated. It is
essential to visualize that this is not so, or rather that such an advantage
is no longer of a strategic nature. Indeed in such a game

vOS) =

hence

if S has

v(S) =

' 3
2
1
0

if S has

'0

1

2

3

elements and 4 does not belong to S f

1

2

elements and 4 belongs to S.
o

4

This is not reduced, as

v((l)) = v((2)) = v((3)) - -1, v((4)) = 3.

If we apply the reduction process of 27.1.4. to this v(S) we find that its
reduced form is

v(S) = 0.

i.e. the game is inessential. (This could have been shown directly by (27 :B)
in 27.4.) Thus this game has a uniquely determined value for each player
1,2, 3, 4: — 1, —1, —1, 3, respectively.

In other words: Player 4’s advantage in this game is one of a fixed
payment (i.e. of cash), and not one of strategic possibilities. The former
is, of course, more definite and tangible than the latter, but of less theoretical
interest since it can be removed by our process of reduction.

299

SPECIAL POINTS IN THE CUBE Q

35 . 1 . 5 . We observed at the beginning of this section that the corners
V, VI, VII differ from I only by permutations of the players. It is easily
verified that the special role of player 4 in I is enjoyed by the players 1,2,3,
in V, VI, VII , respectively.

80.2. The Comer VIII (and II, III, IV). The Three-person Game and a “Dummy”

35 . 2 . 1 . We next consider the games which correspond to the four comers
o : II, III, IV, VIII . As they arise from each other by suitable permutations
of the players 1,2, 3, 4, it suffices to consider one of them, say VIII .

The point VIII corresponds to the values — 1, — 1, — 1 of the coordinates
xi, xt, Xz . Thus the characteristic function v(S) of this game is:

1 0 10

-1 I 1

— 2 12 (and 4 belongs to S)

when S has < elements

2 J 2 (and 4 does not belong to S )

1 ( 3

0 \ 4

(Verification is immediate with the help of (34:1), (34:2), (34:3) in 34.2.1.)
Again, instead of applying to this game the mathematical theory of Chapter
VI, let us first see whether it does not allow an immediate intuitive
interpretation.

The important feature of this game is that the inequality (25:3:c) in
25.3. becomes an equality, i.e. :

(35:7) v(S u T) = w(S) + v(T) if S n T = ©,

when T = (4). That is: If S represents a coalition which does not contain
the player 4, then the addition of 4 to this coalition is of no advantage;
i.e. it does not affect the strategic situation of this coalition nor of its
opponents in any way. This is clearly the meaning of the additivity
expressed by (35:7). 1

36 . 2 . 2 . This circumstance suggests the following conclusion, — which
is of course purely heuristic. 2 Since the accession of player 4 to any

1 Note that the indifference in acquiring the cooperation of 4 is expressed by (35:7),
and not by

y(S U T) - v(jS).

That is, a player is “indifferent” as a partner, not if his accession does not alter the value
of a coalition but if he brings into the coalition exactly the amount which — and no more
than — he is worth outside.

This remark may seem trivial ; but there exists a certain danger of misunderstanding,
particularly in non-reduced games where v((4)) > 0, — i.e. where the accession of 4
(although strategically indifferent!) actually increases the value of a coalition.

Observe also that the indifference of S and T ■* (4) to each other is a strictly recip-
rocal relationship.

* We shall later undertake exact discussion on the basis of 30.1. At that time it will
be found also that all these games are special cases of more general classes of some
importance. (Cf. Chap. IX, particularly 41.2.)

300

ZERO-SUM FOUR-PERSON GAMES

coalition appears to be a matter of complete indifference to both sides, it
seems plausible to assume that player 4 has no part in the transactions that
constitute the strategy of the game. lie is isolated from the others and
the amount which he can get for himself — v(S ) = — 1 — is the actual value
of the game for him. The other players 1,2,3, on the other hand, play
the game strictly among themselves; hence they are playing a three-
person game. The values of the original characteristic function v(S)
which describes the original three-person game are :

(35:6*)

v(0) = 0,

v((l)) = v((2)) = v((3)) = -1,
v((l,2)) = v((l,3)) = v((2,3)) = 2,
v((l,2,3)) = 1,

= (1,2,3) is now the set
of all players.

(Verify this from (35:6).)

At first sight this three-person game represents the oddity that v(/')
(/' is now the set of all players!) is not zero. This, however, is perfectly
reasonable: by eliminating player 4 we transform the game into one which
is not of zero sum; since we assessed player 4 a value — 1, the others retain
together a value 1. We do not yet propose to deal with this situation
systematically. (Cf. footnote 2 on p. 299.) It is obvious, however, that
this condition can be remedied by a slight generalization of the transforma-
tion used in 27.1. We modify the game of 1,2,3 by assuming that each
one got the amount £ in cash in advance, and then compensating for this
by deducting equivalent amounts from the v(S) values in (35:6*). Just
as in 27.1., this cannot affect the strategy of the game, i.e. it produces a
strategically equivalent game. 1

After consideration of the compensations mentioned above 2 we obtain
the new characteristic function:

v'(©) = 0,

v'((l)) = v'((2)) = v'((3)) = - *,

(35:6**) v'((l,2)) = v'((l,3)) = v'((2,3)) = *,

v'((l,2,3)) = 0.

This is the reduced form of the essential zero-sum three-person game dis-
cussed in 32. — except for a difference in unit: We have now y = $ instead

'In the terminology of 27.1.1.: a? = a\ =■ The condition there which

we have infringed is (27:1): ^ a° t = 0. This is necessary since we started with a non-

zero-sum game.

Even ^ aj = 0 could be safeguarded if we included player 4 in our considerations,
»

putting aj =* 1. This would leave him just as isolated as before, but the necessary
compensation would make v((4)) = 0, with results which are obvious.

One can sum this up by saying that in the present situation it is not the reduced
form of the game which provides the best basis of discussion among all strategically
equivalent forms.

* I.e. deduction of as many times J from v(S) as 8 has elements.

301

SPECIAL POINTS IN THE CUBE Q

of the 7 = 1 of (32:1) in 32.1.1. Thus we can apply the heuristic results
of 23.1.3., or the exact results of 32. 1 Let us restrict ourselves, at any
rate, to the solution which appears in both cases and which is the simplest
one: (32:B) of 32.2.3. This is the set of imputations (32:6) in 32.2.1., which
we must multiply by the present value of 7 = 4; i.e.:

( _
1

4 2

s,

It, — 4, 41, If, f, — 4 I •

(The players are, of course, 1,2,3.) In other words: The aim of the strategy
of the players 1,2,3 is to form any coalition of two; a player who succeeds
in this, i.e. who is victorious, gets f , and a player who is defeated gets — f.
Now each of the players 1,2,3 of our original game gets the extra amount
i beyond this, — hence the above amounts f , — 4 must be replaced by
1 , - 1 .

36.2.3. Summing up:

(35 :B) This is a game in which the player 4 is excluded from all

coalitions. The strategic aim of the other players 1,2,3 is to
form any coalition of two. Player 4 gets — 1 at any rate. Any
other player 1,2,3 gets the amount 1 when he is among the win-
ners, and the amount — 1 when he is defeated. All this is based
on heuristic considerations.

One might say more concisely that this four-person game is only an
“ inflated” three-person game: the essential three-person game of the players
1,2,3, inflated by the addition of a “ dummy ” player 4. We shall see later
that this concept is of a more general significance. (Cf. footnote 2 on
p. 299.)

36.2.4. One might compare the dummy role of player 4 in this game
with the exclusion a player undergoes in the discriminatory solution (32: A)
in 32.2.3., as discussed in 33.1.2. There is, however, an important differ-
ence between these two phenomena. In our present set-up, player 4 has
really no contribution to make to any coalition at all; he stands apart by
virtue of the characteristic function v(£). Our heuristic considerations
indicate that he should be excluded from all coalitions in all acceptable
solutions. We shall see in 46.9. that the exact theory establishes just
this. The excluded player of a discriminatory solution in the sense of
33.1.2. is excluded only in the particular situation under consideration. As
far as the characteristic function of that game is concerned, his role is the
same as that of all other players. In other words: The “ dummy” in our
present game is excluded by virtue of the objective facts of the situation
(the characteristic function v(S)). 2 The excluded player in a discrimi-
natory solution is excluded solely by the arbitrary (though stable)
“prejudices” that the particular standard of behavior (solution) expresses.

1 Of course the present discussion is heuristic in any event. As to the exact treat-
ment, cf. footnote 2 on p. 299.

2 This is the il physical background/ 1 in the sense of 4.6.3.

302

ZERO-SUM FOUR-PERSON GAMES

We observed at the beginning of this section that the corners II, III,
IV differ from VIII only by permutations of the players. It is easily
verified that the special role of player 4 in VIII is enjoyed by the players
1,2,3 in II, III , IV, respectively.

35.3. Some Remarks Concerning the Interior of Q

35.3.1. Let us now consider the game which corresponds to the center of
Q, i.e. to the values 0,0,0 of the coordinates x h x 2) x%. This game is clearly
unaffected by any permutation of the players 1,2, 3, 4, i.e. it is symmetric.
Observe that it is the only such game in Q, since total symmetry means
invariance under all permutations of x h x 2 , Xz and sign changes of any two
of them (cf. 34.3.); hence Xi = x 2 = x 3 = 0.

The characteristic function v(S) of this game is:

1 0 / 0

-1 \ 1

0 when S has < 2 elements. 1

1 / 3

0 \ 4

(Verification is immediate with the help of (34:1), (34:2), (34:3) in 34.2.1.)
The exact solutions of this game are numerous; indeed, one must say that
they are of a rather bewildering variety. It has not been possible yet to
order them and to systematize them by a consistent application of the exact
theory, to such an extent as one would desire. Nevertheless the known
specimens give some instructive insight into the ramifications of the theory.
We shall consider them in somewhat more detail in 37. and 38.

At present we make only this (heuristic) remark: The idea of this
(totally) symmetric game is clearly that any majority of the players (i.e. any
coalition of three) wins, whereas in case of a tie (i.e. when two coalitions
form, each consisting of two players) no payments are made.

35.3.2. The center of Q represented the only (totally) symmetric game
in our set-up: with respect to all permutations of the players 1,2, 3, 4. The
geometrical picture suggests consideration of another symmetry as well:
with respect to all permutations of the coordinates X\, x 2 , x *. In this way
we select the points of Q with

(C5:9) xi = x 2 = xz,

which form a main diagonal of Q, the line

(35:10) /- center- VIII .

We saw at the beginning of 34.3.1. that this symmetry means precisely
that the game is invariant with respect to all permutations of the players
1,2,3. In other words:

1 This representation shows once more that the game is symmetric, and uniquely
characterized by this property. Cf. the analysis of 28.2.1.

303

SPECIAL POINTS IN THE CUBE Q

The main diagonal (35:9), (35:10) represents all those games which are
symmetric with respect to the players 1,2,3, i.e. where only player 4 may
have a special role.

Q has three other main diagonals (//-center- F, ///-center-F/, /F- center
F//), and they obviously correspond to those games where another player
(players 1,2,3, respectively) alone may have a special role.

Let us return to the main diagonal (35:9), (35:10). The three games
which we have previously considered (/, VIII , Center) lie on it; indeed in
all these games only player 4 had a special role. 1 Observe that the entire
category of games is a one-parameter variety. Owing to (35:9), such a
game is characterized by the value x\ in

(35:11) -1 £ X! ^ 1.

The three games mentioned above correspond to the extreme values X\ = 1,
X\ = — 1 and to the middle value X\ = 0. In order to get more insight into
the working of the exact theory, it would be desirable to determine exact
solutions for all these values of xi, and then to see how these solutions shift
as X\ varies continuously along (35:10). It would be particularly instruc-
tive to find out how the qualitatively different kinds of solutions recognized
for the special values Xi = —1, 0, 1 go over into each other. In 36.3.2.
we shall give indications about the information that is now available in this
regard.

36 . 3 . 3 . Another question of interest is this: Consider a game, i.e. a
point in Q, where we can form some intuitive picture of what solutions to
expect, e.g. the corner VIII . Then consider a game in the immediate
neighborhood of VIII , i.e. one with only slightly changed values of Xi, x*,
Xz. Now it would be desirable to find exact solutions for these neighboring
games, and to see in what details they differ from the solutions of the
original game, — i.e. how a small distortion of xi, X*, x 3 distorts the solu-
tions. 2 Special cases of this problem will be considered in 36.1.2., and at
the end of 37.1.1., as well as in 38.2.7.

35 . 3 . 4 . So far we have considered games that are represented by points
of Q in more or less special positions. 3 A more general, and possibly more
typical problem arises when the representative point X is somewhere
in the interior of Q, in “ general” position, — i.e. in a position with no
particular distinguishing properties.

Now one might think that a good heuristic lead for the treatment of
the problem in such points is provided by the following consideration. We
have some heuristic insight into the conditions at the corners /-F/// (cf.
35.1. and 35.2.). Any point X of Q is somehow u 1 surrounded” by these
corners; more precisely, it is their center of gravity, if appropriate weights

1 In the center not even he.

* This procedure is familiar in mathematical physics, where it is used in attacking
problems which cannot be solved in their general form for the time being: it is the analysis
of perturbation*.

3 Corners, the center, and entire main diagonals.

304

ZERO-SUM FOUR-PERSON GAMES

are used. Hence one might suspect that the strategy of the games, repre-
sented by X , is in some way a combination of the strategies of the (more
familiar) strategies of the games represented by I-VIII. One might even
hope that this “combination” will in some sense be similar to the formation
of the center of gravity which related X to I-VIII. 1

We shall see in 36.3.2. and in 38.2.5.7. that this is true in limited parts
of Q, but certainly not over all of Q. In fact, in certain interior areas of Q
phenomena occur which are qualitatively different from anything exhibited
by I-VIII. All this goes to show what extreme care must be exercised in
dealing with notions involving strategy, or in making guesses about them.
The mathematical approach is in such an early stage at present that much
more experience will be needed before one can feel any self-assurance in
this respect.

36. Discussion of the Main Diagonals
36.1. The Part Adjacent to the Corner VIII.: Heuristic Discussion

36.1.1. The systematic theory of the four-person game has not yet
advanced so far as to furnish a complete list of solutions for all the games
represented by all points of Q. We are not able to specify even one solution
for every such game. Investigations thus far have succeeded only in
determining solutions (sometimes one, sometimes more) in certain parts of
Q. It is only for the eight corners I-VIII that a demonstrably complete
list of solutions has been established. At the present the parts of Q in
which solutions are known at all form a rather haphazard array of linear,
plane and spatial areas. They are distributed all over Q but do not fill it
out completely.

The exhaustive list of solutions which are known for the corners I-VIII
can easily be established with help of the results of Chapters IX and X,
where these games will be fitted into certain larger divisions of the general
theory. At present we shall restrict ourselves to the casuistic approach

1 Consider two points X = \x\, x 2 , x%\ and Y — \y h y 2) y 9 ) in Q. We may view
these as vectors in L 8 and it is indeed in this sense that the formation of a center of
gravity

tX + (1 - t)Y = {tx i -I- (1 - t)yi, tx 2 + (1 - O 2 / 2 , txi + (1 - 02/a!
is understood. (Cf. (16:A:c) in 16.2.1.)

Now if X =* {*i, x 2 , x$j and Y =■ \y h y 2 , 2/a) define the characteristic functions
v(5) and w(S) in the sense of (34:1)- (34:3) in 34.2.1., then tX + (l — t)Y will give, by
the same algorithm, a characteristic function

u (S) ss tv(S) + (1 — t)w(S).

(It is easy to verify this relationship by inspection of the formulae which we quoted.)
And this same u(&) was introduced as center of gravity of v(S ) and w (S) by (27:10) in
27.6.3.

Thus the considerations of the text are in harmony with those of 27.6. That we
are dealing with centers of gravity of more than two points (eight: I-VIII) instead of only
two, is not essential: the former operation can be obtained by iteration of the latter.

It follows from these remarks that the difficulties which are pointed out in the text
below have a direct bearing on 27.6.3., as was indicated there.

DISCUSSION OF THE MAIN DIAGONALS

305

which consists in describing particular solutions in cases where such are
known. It would scarcely serve the purpose of this exposition to give a
precise account of the momentary state of these investigations 1 and it
would take up an excessive amount of space. We shall only give some
instances which, it is hoped, are reasonably illustrative.

36 . 1 . 2 . We consider first conditions on the main diagonal /-Center- V7/J
in Q near its end at VIII , x\ = z 2 = £3 = — 1 (cf. 35.3.3.), and we shall try

vn in

The diagonal /— Center —VIII redrawn
VIII Center 1

-1 0 I

Figure 63.

to extend over the Xi = £2 = £3 > — 1 as far as possible. (Cf. Figure 63.)
On this diagonal

1 0 /0

-1 l 1

2xi j 2 (and 4 belongs to S )

when S has \ elements

— 2xi 12 (and 4 does not belong to S )

1 ( 3

0 \ 4

(Observe that this gives (35:1) in 35.1.1. for x\ = 1 and (35:6) in 35.2.1. for

X\ = —1.) We assume that x\ > — 1 but not by too much, — just how
much excess is to be permitted will emerge later. Let us first consider this
situation heuristically.

Since X\ is supposed to be not very far from —1, the discussion of 35.2.
may still give some guidance. A coalition of two players from among

1 This will be done by one of us in subsequent mathematical publications.

306

ZERO-SUM FOUR-PERSON GAMES

1,2,3 may even now be the most important strategic aim, but it is no longer
the only one: the formula (35:7) of 35.2.1. is not true, but instead

(36:2) v(S u T) > v(S) + v(T) if S n T = ©

when T = (4). 1 Indeed, it is easily verified from (36:1) that this excess
is always 2 * 4 * 2(1 + £i). For X\ = — 1 this vanishes, but we have x\ slightly

> — 1, so the expression is slightly > 0. Observe that for the preceding
coalition of two players other than player 4, the excess in (36:2) 8 is by
(36:1) always 2(1 — Xi). For x x = — 1 this is 4, and as we have Xi slightly

> —1, it will be only slightly < 4.

Thus the first coalition (between two players, other than player 4),
is of a much stronger texture than any other (where player 4 enters into the
picture), — but the latter cannot be disregarded nevertheless. Since the
first coalition is the stronger one, it may be suspected that it will form first
and that once it is formed it will act as one player in its dealings with the
two others. Then some kind of a three-person game may be expected to
take place for the final crystallization.

36 . 1 . 3 . Taking, e.g. (1,2) for this “first” coalition, the surmised three-
person game is between the players (1,2), 3,4. 4 In this game the a, b , c of
23.1. are a = v((3,4)) = 2x h b = v((l,2,4)) = 1, c = v((l,2,3)) = l. 6 *
Hence, if we may apply the results obtained there (all of this is extremely

heuristic!) the player (1,2) gets the amount a = — a — ^ - — - = 1 — X\,

if successful (in joining the last coalition), and —a = — 2x\ if defeated.

player 3 gets the amount 0 — ■ - = X\ if successful, and — b ■

£

The

-1

if defeated. The player 4 gets the amount y

a + b — c
2

Xi if success-

ful, and — c = —1 if defeated.

Since “first” coalitions (1,3), (2,3) may form, just as well as (1,2),
there are the same heuristic reasons as in the first discussion of the three-
person game (in 21.-22.) to expect that the partners of these coalitions will
split even. Thus, when such a coalition is successful (cf. above), its

1 — x i

members may be expected to get — — each, and when it is defeated

xi each.

36 . 1 . 4 . Summing up: If these surmises prove correct, the situation is as
follows:

1 Unless S = © or — T, in which case there is always « in (36:2). I.e. in the pres-
ent situation S must have one or two elements.

2 By footnote 1 above, S has one or two elements and it does not contain 4.

1 I.e., now S , T are two one-element sets, not containing player 4.

4 One might say that (1,2) is a juridical person, while 3,4 are, in our picture, natural

persons.

• In all the formulae which follow, remember that X\ is near to — 1, — i.e. presumably

negative; hence — x\ is a gain, and X\ is a loss.

DISCUSSION OF THE MAIN DIAGONALS

307

If the “first” coalition is (1,2), and if it is successful in finding an ally,
and if the player who joins it in the final coalition is player 3, then the

players 1,2, 3, 4 get the amounts — ^ — , — 2 — ' Xl ’ 1 respectively. If the

player who joins the final coalition is player 4, then these amounts are

replaced by ^ — 1, x,. If the “first” coalition (1,2) is not

successful, i.e. if the players 3,4 combine against it, then the players get the
amounts — x u — xi, x h x\ respectively.

If the “first” coalition is (1,3) or (2,3), then the corresponding permuta-
tion of the players 1,2,3 must be applied to the above.

36.2. The Part Adjacent to the Corner VIII.: Exact Discussion

36 . 2 . 1 . It is now necessary to submit all this to an exact check. The
heuristic suggestion manifestly corresponds to the following surmise:

Let V be the set of these imputations :

(36:3)

1

- Xi
2 ; Xl *

1

— Xl 1 — Xl
~2 ' 2 '

- 1 , Xi

a

ftt

{ -Xl, -Xl, Xl, Xl}

and the imputations which
originate from these by per-
muting the players, (i.e. the
components) 1,2,3.

(Cf. footnote 5, p. 306.) We expect that this V is a solution in the rigorous
sense of 30.1. if x 2 is near to — 1 and we must determine whether this is so>
and precisely in what interval of the Xi.

This determination, if carried out, yields the following result:

(36: A) The set V of (36:3) is a solution if and only if

— 1 ^ xi ^ — i.

This then is the answer to the question, how far (from the starting point
xi = —1, i.e. the corner VIII) the above heuristic consideration guides to
a correct result. 1

36 . 2 . 2 . The proof of (36: A) can be carried out rigorously without any
significant technical difficulty. It consists of a rather mechanical disposal

1 We wish to emphasize that (36: A) does not assert that V is (in the specified range of
Xi) the only solution of the game in question. However, attempts with numerous
similarly built sets failed to disclose further solutions for X\ £ — J (i.e. in the range of
(36:A)). For X\ slightly > — i (i.e. slightly outside the range of (36:A)), where the
V of (36: A) is no longer a solution, the same is true for the solution which replaces it.
Cf. (36 :B) in 36.3.1.

We do not question, of course, that other solutions of the “ discriminatory ” type,
as repeatedly discussed before, always exist. But they are fundamentally different from
the finite solutions V which are now under consideration.

These are the arguments which seem to justify our view that some qualitative change
in the nature of the solutions occurs at

Xi * — £ (on the diagonal /-center- VIII).

308

ZERO-SUM FOUR-PERSON GAMES

of a series of special cases, and does not contribute anything to the clarifica-
tion of any question of principle. 1 The reader may therefore omit reading
it if he feels so disposed, without losing the connection with the main course
of the exposition. He should remember only the statement of the results
in (36 :A).

Nevertheless we give the proof in full for the following reason: The set
V of ( 36 : 3 ) was found by heuristic considerations, i.e. without using the
exact theory of 30.1. at all. The rigorous proof to be given is based on
30 . 1 . alone, and thereby brings us back to the only ultimately satisfactory
standpoint, that of the exact theory. The heuristic considerations were
only a device to guess the solution, for want of any better technique; and
it is a fortunate feature of the exact theory that its solutions can occasionally
be guessed in this way. But such a guess must afterwards be justified
by the exact method, or rather that method must be used to determine in
what domain (of the parameters involved) the guess was admissible.

We give the exact proof in order to enable the reader to contrast and
to compare explicitly these two procedures, — the heuristic and the rigorous.
36 . 2 . 3 . The proof is as follows:

If x\ = — 1, then we are in the corner VIII, and the V of (36:3) coincides
with the set which we introduced heuristically (as a solution) in 35.2.3.,
and which can easily be justified rigorously (cf. also footnote 2 on p. 299).
Therefore we disregard this case now, and assume that

(36:4) X! > -1.

We must first establish which sets SqI = (1,2, 3, 4) are certainly
necessary or certainly unnecessary in the sense of 31.1.2. — since we are
carrying out a proof which is precisely of the type considered there.

The following observations are immediate:

(36:5) By virtue of (31 :H) in 31.1.5., three-element sets S are cer-

tainly necessary, two-element sets are dubious, and all other
sets are certainly unnecessary. 2

(36:6) Whenever a two-element set turns out to be certainly neces-

sary, we may disregard all those three-element sets of which it is
a subset, owing to (31 :C) in 31.1.3.

Consequently we shall now examine the two-element sets. This of course

— >

must be done for all the a in the set V of (36:3).

1 The reader may contrast this proof with some given in connection with the theory of
the zero-sum two-person game, e.g. the combination of 16.4. with 17.6. Such a proof is
more transparent, it usually covers more ground, and gives some qualitative elucidation
of the subject and its relation to other parts of mathematics. In some later parts of this
theory such proofs have been found, e.g. in 46. But much of it is still in the primitive
and technically unsatisfactory state of which the considerations which follow are typical.

* This is due to n — 4.

DISCUSSION OF THE MAIN DIAGONALS 309

Consider first those two-element sets S which occur in conjunction
— >

with a '. l As a[ = —1 we may exclude by (31 :A) in 31.1.3. the possibility
that S contains 4. S = (1,2) would be effective if a[ + a' i g t v((l,2)),
i.e. 1 — Xi ^ — 2xi, *i sS — 1 which is not the case by (36:4). S = (1,3)

is effective if a[ + v((l,3)), i.e. 1 * Xl £ -2x,, *, g - i Thus

the condition

(36:7) *1 S - *

which we assume to be satisfied makes its first appearance. S = (2,3) we

do not need, since 1 and 2 play the same role in a ' (cf. footnote 1 above).

We now pass to a ". As a' 3 ' = ~lwe now exclude the S which contains 3

(cf. above). S = (1,2) is disposed of as before, since a ' and a " agree in
these components. S = (1,4) would be effective if a" + a" ^ v((l,4)),

1 “I - Xl

i e. — g ^ 2xi, Xi ^ $ which, by (36:7), is not the case. S = (2,4) is
discarded in the same way.

Finally we take a'". S = (1,2) is effective: a" + a'^' = v((l,2))
i.e. — 2xi = — 2xi. S = (1,3) need not be considered for the following

reason: We are already considering S = (1,2) for a if we interchange 2
and 3 (cf. footnote 1 above) this goes over into (1,3), with the components

-xi, -xi. Our original S = (1,3) for a'” with the components -x x , Xi
is thus rendered unnecessary by (31 :B) in 31.1.3., as — Xi ^ xi owing to
(36:7). S = (2,3) is discarded in the same way. S = (1,4) would be
effective if «i" + a'" g v((l,4)) i.e. 0 g 2x,, xi ^ 0, which, by (36:7), is
not the case. S = (2,4) is discarded in the same way. S = (3,4) is effec-
tive: a' t " + a'" = v((3,4)), i.e. 2x x = 2xi.

Summing up:

(36:8) Among the two-element sets S the three given below are

certainly necessary, and all others are certainly unnecessary:

(1,3) for (1,2) and (3,4) for 7'".

Concerning three-element sets S: By (31: A) in 31.1.3. we may exclude
— * — *

those containing 4 for a ' and 3 for a Consequently only (1,2,3) is left
for a' and (1,2,4) for a". Of these the former is excluded by (36:6),
as it contains the set (1,3) of (36:8). For a'" every three-element set

1 Here, and in the entire discussion which follows, we shall make use of the freedom to
apply permutations of 1,2,3 as stated in (36:3), in order to abbreviate the argumentation.
Hence the reader must afterwards apply these permutations of 1,2,3 to our results.

310 ZERO-SUM FOUR-PERSON GAMES

contains the set (1,2) or the set (3,4) of (36:8); hence we may exclude it by
(36:6).

Summing up:

(36:9) Among the three-element sets S, the one given below is

certainly necessary, and all others are certainly unnecessary: 1

(1,2,4) for a".

— ►

36.2.4. We now verify (30:5:a) in 30.1.1., i.e. that no a of V dominates
any 0 of V.

a = a By (36:8), (36:9) we must use S = (1,3). Can a ' dominate

with this S any 1,2,3 permutation of a ' or a " or a '"? This requires first

the existence of a component < X\ (this is the 3 component of a ') among

the 1,2,3 components of the imputation in question. Thus a ' and a
— ►

are excluded. 2 Even in a " the 1,2 components are excluded (cf. footnote 2)
but the 3 component will do. But now another one of the 1,2,3 components

of this imputation a " must be < — 0 Xl (this is the 1 component of a '),

and this is not the case; the 1,2 components of a " are both =

1 ~ Xi

a = a": By (36:8), (36:9) we must use S = (1,2,4). Can a "dominate

with this S any 1,2,3 permutation of a ' or a " or a "'? This requires first
that the 4 component of the imputation in question be < X\ (this is the

4 component of a"). Thus a "and a are excluded. For a 'we must

1 — Xi

require further that two of its 1,2,3 components be < — ^ is the 1
as well as the 2 component of a "), and this is not the case; only one of these

components is s*

1 - Xi

a = a'": By (36:8), (36:9) we must use S = (1,2) and then S = (3,4).

S = (1,2) : Can a!" dominate with this S as described above? This requires
the existence of two components < —xi (this is the 1 as well as the 2 com-
— ►

ponent of a'") among the 1,2,3 components of the imputation in question.

This is not the case for a'", as only one of these components is ^ —Xi

1 As every three-element set is certainly necessary by (36:5) above, this is another
instance of the phenomenon mentioned at the end of footnote 1 on p. 274.

1 — Xi 1

1 Indeed — ^ — £ Xi, i.e. zi £ £ and -Xi £ *i, i.e. Xi £ 0 — both by (36:7).

DISCUSSION OF THE MAIN DIAGONALS

311

there. Nor is it the case for a ' or a ", as only one of those components is
there. 1 S = (3,4): Can a "' dominate with this S as described

1

Xi

above? This requires first that the 4 component of the imputation in

question be < Xi (this is the 4 component of a "'). Thus a " and a '" are
— ►

excluded. For a ' we must require further the existence of a component

< x\ (this is the 3 component of a "') among its 1,2,3 components, and
this is not the case; all these components are ^ X\ (cf. footnote 2 on p. 310).
This completes the verification of (30:5:a).

36 . 2 . 5 . We verify next (30:5:b) in 30. 1. 1 . , i.e. that an imputation 0 which

is undominated by the elements of V must belong to V.

— >

Consider a 0 undominated by the elements of V. Assume first that
p A < Xi. If any one of p i, 0 2 , 0 3 were < x h we could make (by permuting

1,2,3) 0 3 < X\. This gives a h p with S = (3,4) of (36:8). Hence

Pu P2, 03 ^ Xi.

1

If any two of P i, 0 2 , 0 3 were < 2

1 - X!

Xi

, we could make (by permuting 1,2.3)

P\> P 2 <

This gives a " H p with S — (1,2,4) of (36:9). Hence,

at most one of 0i, p 2} 0 3 is < 1 i.e. two are ^ * - 0 — • By permuting

1,2,3, we can thus make

Pu 0* ^

1 — X\

Clearly pi ^ — 1. Thus each component of P is ^ the corresponding com-
— ►

ponent of a and since both are imputations 2 it follows that they coincide:
P = a and so it is in V.

Assume next that 0 4 ^ x\. If any two of p h 0 2 , p 8 were < — Xi, we

could make (by permuting 1,2,3) Pi, 02 < — xi. This gives a"' H p with
S = (1,2) of (36:8). Hence, at most one of Pi, p 2 , 0 a is < — Xi, i.e. two are
^ — xi. By permuting 1,2,3 we can make

Ph 02 S -xi.

— >

If 0s ^ Xi, then all this implies that each component of 0 is ^ the cor-

1 - Xi

1 And

2

^ -*i, i.e. xi £ -1.

1 Consequently for both the sum of all components is the same: zero.

312

ZERO-SUM FOUR-PERSON GAMES

responding component of a and since both are imputations (cf . footnote 2,

on p. 311) it follows that they coincide: 0 = a and so it is in V.

Assume therefore that 0 8 < £i. If any one of 0i, 02 were <

Xi

we could make (by permuting 1,2) 0i <
with S = (1,3) of (36:8). Hence

01 , 02 £

1

Xi

This gives a ' H 0

Clearly 03 ^ —1. Thus each component of 0 is ^ the corresponding com-
ponent of a", and since both are imputations (cf. footnote 2, p. 311), it

follows that they coincide: 0 = a ", and so it is in V.

This completes the verification of (30 :5 b). 1
So we have established the criterion (36: A). 2

36.3. Other Parts of the Main Diagonals

36.3.1. When Xi passes outside the domain (36 :A) of 36.2.1., i.e. when
it crosses its border at Xi = — i, then the V of (36:3) id. ceases to be a
solution. It is actually possible to find a solution which is valid for a certain
domain in x x > — i (adjoining xi = — £), which obtains by adding to the
V of (36:3) the further imputations

(36:10)

1 — Xi

-> -3l,

■ 1 + Xi

, *1

and permutations as in
(36:3). 3

The exact statement is actually this:

(36 :B) The set V of (36:3) and (36:10) is a solution if and only if

- i < xi ^ 0. 4

1 The reader will observe that in the course of this analysis all sets of (36:8), (36:9)

— ► — > — ►

have been used for dominations, and 0 had to be equated successively to all three a ', a ",

a of (36:3).

* Concerning xi = — 1, cf. the remarks made at the beginning of this proof.

3 An inspection of the above proof shows that when X\ becomes > — J, this goes

wrong: The set S — (1,3) (and with it (2,3)) is no longer effective for a '. Of course this
rehabilitates the three-element set S «■ (1,2,3) which was excluded solely because (1,3)
(and (2,3)) is contained in it.

— y

.Thus domination by this element of V, a now becomes more difficult, and it is
therefore not surprising that an increase of the set V must be considered in the search
for a solution.

4 Observe the discontinuity at x\ - — $ which belongs to (36:A) and not to (36:B)I
The exact theory is quite unambiguous, even in such matters.

THE CENTER AND ITS ENVIRONS

313

The proof of (36 :B) is of the same type as that of (36: A) given above,
and we do not propose to discuss it here.

The domains (36 :A) and (36 :B) exhaust the part Xi ^ 0 of the entire
available interval — 1 ^ x x g 1 — i.e. the half VIII - Center of the diagonal
V/JJ-Center-J.

36.3.2. Solutions of a nature similar to V described in (36 :A) of 36.2.1.
and in (36 :B) of 36.3.1., have been found on the other side xi > 0 — i.e.
the half Center-/ of the diagonal — as well. It turns out that on this half,
qualitative changes occur of the same sort as in the first half covered by
(36: A) and (36 :B). Actually three such intervals exist, namely:

(36 :C) 0 ^ Xl < i,

(36 :D) i < xi g i,

(36 :E) i g xi S 1.

(Cf. Figure 64, which is to be compared with Figure 63.)

VIII Carter J

Figure 64.

We shall not discuss the solutions pertaining to (36 :C), (36 :D), (36 :E). 1

The reader may however observe this: x\ = 0 appears as belonging
to both (neighboring) domains (36 :B) and (36 :C), and similarly Xi = i to
both domains (36 :D) and (36 :E). This is so because, as a close inspection
of the corresponding solutions V shows that while qualitative changes in the
nature of V occur at z t = 0 and £, these changes are not discontinuous.

The point X\ — i, on the other hand, belongs to neither neighboring
domain (36 :C) or (36 :D). It turns out that the types of solutions V which
are valid in these two domains are both unusable at Xi = i. Indeed, the
conditions at this point have not been sufficiently clarified thus far.

37. The Center and Its Environs

37.1. First Orientation Concerning the Conditions around the Center

37.1.1. The considerations of the last section were restricted to a one-
dimensional subset of the cube Q: The diagonal VIII-center-I. By using
the permutations of the players 1,2, 3, 4, as described in 34.3., this can be
made to dispose of all four main diagonals of Q. By techniques that are
similar to those of the last section, solutions can also be found along some
other one-dimensional lines in Q. Thus there is quite an extensive net of
lines in Q on which solutions are known. We do not propose to enumerate
them, particularly because the information that is available now corresponds
probably to only a transient state of affairs.

1 Another family of solutions, which also cover part of the same territory, will be
discussed in 38.2. Cf. in particular 38.2.7., and footnote 2 on p. 328.

314

ZERO-SUM FOUR-PERSON GAMES

This, however, should be said : such a search for solutions along isolated
one-dimensional lines, when the whole three-dimensional body of the cube Q
waits for elucidation, cannot be more than a first approach to the problem.
If we can find a three-dimensional part of the cube — even if it is a small
one — for all points of which the same qualitative type of solutions can be
used, we shall have some idea of the conditions which are to be expected.
Now such a three-dimensional part exists around the center of Q. For this
reason we shall discuss the conditions at the center.

37 . 1 . 2 . The center corresponds to the values 0,0,0 of the coordinates
X\, x z and represents, as pointed out in 35.3.1., the only (totally) sym-
metric game in our set-up. The characteristic function of this game is:

1 0
1

2 elements.

3
4

(Cf. (35:8) id.) As in the corresponding cases in 35.1., 35.2., 36.1., we begin
again with a heuristic analysis.

This game is obviously one in which the purpose of all strategic efforts
is to form a three-person coalition. A player who is left alone is clearly a
loser, any coalition of 3 in the same sense a winner, and if the game should
terminate with two coalitions of two players each facing each other, then
this must obviously be interpreted as a tie.

The qualitative question which arises here is this: The aim in this
game is to form a coalition of three. It is probable that in the negotiations
which precede the play a coalition of two will be formed first. This coalition
will then negotiate with the two remaining players, trying to secure the
cooperation of one of them against the other. In securing the adherence
of this third player, it seems questionable whether he will be admitted into
the final coalition on the same conditions as the two original members.
If the answer is affirmative, then the total proceeds of the final coalition, 1,
will be divided equally among the three participants: i, £, *. If it is
negative, then the two original members (belonging to the first coalition
of two) will probably both get the same amount, but more than Thus 1
will be divided somewhat like this: i + c, i + e, i — 2c with an e > 0.

37 . 1 . 3 . The first alternative would be similar to the one which we
encountered in the analysis of the point I in 35.1. Here the coalition
(1,2,3), if it forms at all, contains its three participants on equal terms.
The second alternative corresponds to the situation in the interval analyzed
in 36. 1.-2. Here any two players (neither of them being player 4) combined
first, and this coalition then admitted either one of the two remaining play-
ers on less favorable terms.

37 . 1 . 4 . The present situation is not a perfect analogue of either of these.

(37:1)

v(S) = <

0

-1

0

1

0

THE CENTER AND ITS ENVIRONS

315

In the first case the coalition (1,2) could not make stiff terms to player 3
because they absolutely needed him: if 3 combined with 4, then 1 and 2
would be completely defeated; and (1,2) could not, as a coalition, combine
with 4 against 3, since 4 needed only one of them to be victorious (cf. the
description in 35.1.3.). In our present game this is not so: the coalition
(1,2) can use 3 as well as 4, and even if 3 and 4 combine against it, only a
tie results.

In the second case the discrimination against the member who joins
the coalition of three participants last is plausible, since the original coali-
tion of two is of a much stronger texture than the final coalition of three.
Indeed, as X\ tends to — 1, the latter coalition tends to become worthless; cf.
the remarks at the end of 36.1.2. In our present game no such qualitative
difference can be recognized: the first coalition (of two) accounts for the
difference between defeat and tie, while formation of the final coalition (of
three) decides between tie and victory.

We have no satisfactory basis for a decision except to try both alterna-
tives. Before we do this, however, an important limitation of our consider-
ations deserves attention.

37.2. The Two Alternatives and the Role of Symmetry

37.2.1. It will be noted that we assume that the same one of the two
alternatives above holds for all four coalitions of three players. Indeed, we
are now looking for symmetric solutions only, i.e. solutions which contain,

along with an imputation a = (ou, a 2 , ou}, all its permutations.

Now a symmetry of the game by no means implies in general the
corresponding symmetry in each one of its solutions. The discriminatory
solutions discussed in 33.1.1. make this clear already for the three-person
game. We shall find in 37.6. further instances of this for the symmetric
four-person game now under consideration.

It must be expected, however, that asymmetric solutions for a symmetric
game are of too recondite a character to be discovered by a first heuristic
survey like the present one. (Cf. the analogous occurrence in the three-
person game, referred to above.) This then is our excuse for looking at
present only for symmetric solutions.

37.2.2. One more thing ought to be said: it is not inconceivable that,
while asymmetric solutions exist, general organizational principles, like
those corresponding to our above two alternatives, are valid either for the
totality of all participants or not at all. This surmise gains some strength
from the consideration that the number of participants is still very low, and
may actually be too low to permit the formation of several groups of par-
ticipants with different principles of organization. Indeed, we have only
four participants, and ample evidence that three is the minimum number
for any kind of organization. These somewhat vague considerations will
find exact corroboration in at least one special instance in (43 :L) et seq. of

43.4.2. For the present case, however, we are not able to support them by
any rigorous proof.

316

ZERO-SUM FOUR-PERSON GAMES

37.3. The First Alternative at the Center

37.3.1. Let us now consider the two alternatives of 37.1.2. We take
them up in reverse order.

Assume first that the two original participants admit the third one
under much less favorable conditions. Then the first coalition (of two)
must be considered as the core on which the final coalition (of three) crystal-
lizes. In this last phase the first coalition must therefore be expected to
act as one player in its dealings with the two others, thus bringing about
something like a three-person game. If this view is sound, then we may
repeat the corresponding considerations of 36.1.3.

Taking, e.g. (1,2), as the “first” coalition, the surmised three-person
game is between the players (1, 2), 3,4. The considerations referred to
above therefore apply literally, only with changed numerical values: a = 0,
b = c = 1 and so a = 1, = y = 0. 1

Since the “first” coalition may consist of any two players, there are
heuristic reasons similar to those in the discussion of the three-person game
(in 21.-22.) to expect that the partners in it will split even : when an ally is
found, as well as when a tie results, the amount to be divided being 1 or 0
respectively. 2

37.3.2. Summing up : if the above surmises prove correct, the situation is
as follows:

If the “first” coalition is (1,2) and if it is successful in finding an
ally, and if the player who joins it in the final coalition is 3, then the players
1,2, 3, 4 get the amounts *, *, 0, —1 respectively. If the “first” coalition
is not successful, i.e. if a tie results, then these amounts are replaced by
0 , 0 , 0 , 0 .

If the distribution of the players is different, then the corresponding
permutation of the players 1,2, 3, 4 must be applied to the above.

It is now necessary to submit all this to an exact check. The heuristic
suggestion manifestly corresponds to the following surmise:

Let V be the set of these following imputations

i> 0, —1} and the imputations which originate from

^ ' "a ''={0000) these by permuting the players (i.e. the

components) 1,2, 3, 4.

We expect that this V is a solution.

1 The essential difference between this discussion and that referred to, is that player 4
is no longer excluded from the “ first” coalition.

* The argument in this case is considerably weaker than in the case referred to (or in
the corresponding application in 36.1.3.) since every u first” coalition may now wind up
in two different ways (tie or victory). The only satisfactory decision as to the value of
the argumentation obtains when the exact theory is applied. The desired justification is
actually contained in the proof of 38.2. 1.-3.; indeed, it is the special case

2 /i * yi - y% - 2/4 - l

of (38:D) in 38.2.3.

THE CENTER AND ITS ENVIRONS

317

A rigorous consideration, of the same type as that which constitutes

36.2., shows that this V is indeed a solution in the sense of 30.1. We do
not give it here, particularly because it is contained in a more general proof
which will be given later. (Cf. the reference of footnote 2 on p. 316.)

37.4. The Second Alternative at the Center

37.4.1. Assume next that the final coalition of three contains all its
participants on equal terms. Then if this coalition is, say (1,2,3), the players
1,2, 3, 4 get the amounts — 1 respectively.

It would be rash to conclude from this that we expect the set of impu-
tations V to which this leads, to be a solution; i.e. the set of these imputa-
— >

tions a = {ai, « 2 , « 3 , oti } :

(37:3) a = {£, i, £, —1} and permutations as in (37:2).

We have made no attempt as yet to understand how this formation of the
final coalition in “one piece” comes about, without assuming the previous
existence of a favored two-person core.

37.4.2. In the previous solution of (37 :2) such an explanation is discern-
able. The stratified form of the final coalition is expressed by the imputa-
tion a ' and the motive for just this scheme of distribution lies in the threat

— > — >

of a tie, expressed by the imputation a ". To put it exactly: the a ' form a

— ►

solution only in conjunction with the a ", and not by themselves.

In (37:3) this second element is lacking. A direct check in the sense

of 30.1. discloses that the a fulfill condition (30:5 :a) there, but not (30:5 :b).
I.e. they do not dominate each other, but they leave certain other imputa-
tions undominated. Hence further elements must be added to V. 1

— ►

This addition can certainly not be the a" = {0,0,0,0J of (37:2) since

that imputation happens to be dominated by a '". 2 In other words the

extension (i.e. stabilization, in the sense of 4.3.3.) of a to a solution must
be achieved by entirely different imputations (i.e. threats) in the case of

the a of (37:3) as in the case of the a ' of (37:2).

It seems very difficult to find a heuristic motivation for the steps which
are now necessary. Luckily, however, a rigorous procedure is possible
from here on, thus rendering further heuristic considerations unnecessary.

1 To avoid misunderstandings : It is by no means generally true that any set of imputa-
tions which do not dominate each other can be extended to a solution. Indeed, the
problem of recognizing a given set of imputations as being a subset of some (unknown)
solution is still unsolved. Cf. 30.3.7.

In the present case we are just expressing the hope that such an extension will prove
possible for the V of (37:3), and this hope will be further justified below.

* With S - (1,2,3).

318

ZERO-SUM FOUR-PERSON GAMES

Indeed, one can prove rigorously that there exists one and only one sym-
metric extension of the V of (37 :3) to a solution. This is the addition of

these imputations a = {ai, a 2 , as, a 4 }

(37:4)

- {*,*,-*,

and permutations as in (37:2).

37 . 4 . 3 . If a common-sense interpretation of this solution, i.e. of its

constituent a /v of (37 :4), is wanted, it must be said that it does not seem to

be a tie at all (like the corresponding a " of (37 :2)) — rather, it seems to be
some kind of compromise between a part (two members) of a possible
victorious coalition and the other two players. However, as stated above,
we do not attempt to find a full heuristic interpretation for the V of (37 :3)
and (37 :4) ; indeed it may well be that this part of the exact theory is already
beyond such possibilities. 1 Besides, some subsequent examples will
illustrate the peculiarities of this solution on a much wider basis. Again
we refrain from giving the exact proof referred to above.

37.5. Comparison of the Two Central Solutions

37 . 6 . 1 . The two solutions (37:2) and (37:3), (37:4), which we found
for the game representing the center, present a new instance of a possible
multiplicity of solutions. Of course we had observed this phenomenon
before, namely in the case of the essential three-person game in 33.1.1. But
there all solutions but one were in some way abnormal (we described this by
terming them “ discriminatory ”). Only one solution in that case was a
finite set of imputations; that solution alone possessed the same symmetry
as the game itself (i.e. was symmetric with respect to all players). This
time conditions are quite different. We have found two solutions which are
both finite sets of imputations, 2 and which possess the full symmetry of the
game. The discussion of 37.1.2. shows that it is difficult to consider either
solution as “abnormal” or “discriminatory” in any sense; they are distin-
guished essentially by the way in which the accession of the last participant
to the coalition of three is treated, and therefore seem to correspond to two
perfectly normal principles of social organization.

37 . 6 . 2 . If anything, the solution (37:3), (37:4) may seem the less normal

one. Both in (37:2) and in (37:3), (37:4) the character of the solution was

— ►

determined by those imputations which described a complete decision, a '
and a'” respectively. To these the extra “stabilizing” imputations, a”
and a /v , had to be added. Now in the first solution this extra a " had an

1 This is, of course, a well known occurrence in mathematical-physical theories, even
if they originate in heuristic considerations.

* An easy count of the imputations given and of their different permutations shows
that the solution (37:2) consists of 13 elements, and the solution (37:3), (37:4) of 10.

THE CENTER AND ITS ENVIRONS 319

obvious heuristic interpretation as a tie, while in the second solution the

nature of the extra a IV appeared to be more complex.

A more thorough analysis discloses, however, that the first solution
is surrounded by some peculiar phenomena which can neither be explained
nor foreseen by the heuristic procedure which provided easy access to this
solution.

These phenomena are quite instructive from a general point of view too,
because they illustrate in a rather striking way some possibilities and
interpretations of our theory. We shall therefore analyze them in some
detail in what follows. We add that a similar expansion of the second
solution has not been found up to now.

37.6. Un symmetric al Central Solutions

37 . 6 . 1 . To begin with, there exist some finite but asymmetrical solutions
which are closely related to (37 :2) in 37.3.2. because they contain some of the
imputations {i, i, 0, — 1 ) . l One of these solutions is the one which obtains
when we approach the center along the diagonal /-Center- V/// from either
side, and use there the solutions referred to in 36.3. I.e. : it obtains by
continuous fit to the domains (36 :B) and (36 :C) there mentioned. (It
will be remembered that the point X\ = 0, i.e. the center, belongs to both
these domains, cf. 36.3.2.) Since this solution can be taken also to express
a sui generis principle of social organization, we shall describe it briefly.

This solution possesses the same symmetry as those which belong to the
games on the diagonal /-Center- VIII y as it is actually one of them : symmetric
with respect to players 1,2,3, while player 4 occupies a special position. 2
We shall therefore describe it in the same way we did the solutions on
the diagonal, e.g. in (36:3) in 36.2.1. Here only permutations of the
players 1,2,3 are suppressed, while in the descriptions of (37 :3) and (37 :4) we
suppressed all permutations of the players 1,2, 3, 4.

37 . 6 . 2 . For the sake of a better comparison, we restate with this notation
(i.e. allowing for permutations of 1,2,3 only) the definition of our first fully-
symmetric solution (37:2) in 37.3.2. It consists of these imputations:*

fi' = {*,*, 0 , - 1 )

7 " - {*,*, -1,0}

7 m = {*,o, -i,ij
7 ,v = {0,0,0,01

and the imputations which originate from
these by permuting the players 1,2,3.

1 I.e. some but not all of the 12 permutations of this imputation.

* That the position of player 4 in the solution is really different from that of the others,
is what distinguishes this solution from the two symmetric ones mentioned before.

* Our 0 ', 0 ", & "' exhaust the a ' of (37:2) in 37.3.2., while & lv is a ", id.

— > — — > — ►

a ' had to be represented by the three imputations 0 ', 0 ", 0 '" because this
system of representation makes it necessary to state in which one of the three possible
positions of that imputation (i.e. the values J, 0, —1) the player 4 is found.

320

ZERO-SUM FOUR-PERSON GAMES

Now the (asymmetric) solution to which we refer consists of these
imputations:

“a" liiv qq in ^7-9*^ and the imputations which origi-
(37:5) P ' P \ • ) natefrom these by permuting the

P v = {i, 0, - i, 01 1 * * players 1,2,3.

Once more we omit giving the proof that (37 :5) is a solution. Instead
we shall suggest an interpretation of the difference between this solution
and that of (37:2) — i.e. of the first (symmetric) solution in 37.3.2.

37 . 6 . 3 . This difference consists in replacing

- {*, 0 , - 1 ,*}

by

7 V = {*, 0 , -*, 0 }.

That is: the imputation f$ — in which the player 4 would belong to the
“first” coalition (cf. 37.3.1.), i.e. to the group which wins the maximum

amount £ — is removed, and replaced by another imputation p r . Player 4

now gets somewhat less and the losing player among 1,2,3 (in this arrange-

— >

ment player 3) gets somewhat more than in P This difference is pre-
cisely i , so that player 4 is reduced to the tie position 0, and player 3 moves
from the completely defeated position — 1 to an intermediate position — $.

Thus players 1,2,3 form a “privileged” group and no one from the out-
side will be admitted to the “first” coalition. But even among the three
members of the privileged group the wrangle for coalition goes on, since
the “first” coalition has room for two participants only. It is worth noting
that a member of the privileged group may even be completely defeated, as

in P ", — but only by a majority of his “ class ” who form the “ first ” coalition
and who may admit the “unprivileged” player 4 to the third membership
of the “final” coalition, to which he is eligible.

37 . 6 . 4 . The reader will note that this describes a perfectly possible form
of social organization. This form is discriminatory to be sure, although
not in the simple way of the “discriminatory” solutions of the three-person
game. It describes a more complex and a more delicate type of social
inter-relation, due to the solution rather than to the game itself. 2 One
may think it somewhat arbitrary, but since we are considering a “society”
of very small size, all possible standards of behavior must be adjusted rather
precisely and delicately to the narrowness of its possibilities.

We scarcely need to elaborate the fact that similar discrimination
against any other player (1,2,3 instead of 4) could be expressed by suitable

1 This imputation 0 v is reminiscent in its arrangement of a IV in (37:4) of 37.4.2.,

but it has not been possible to make anything of that analogy.

* As to this feature, cf. the discussion of 85.2.4.

NEIGHBORHOOD OF THE CENTER

321

solutions, which could then be associated with the three other diagonals
of the cube Q.

38. A Family of Solutions for a Neighborhood of the Center
38.1. Transformation of the Solution Belonging to the First Alternative at the Center

38.1.1. We continue the analysis of the ramifications of solution (37:2)
in 37.3.2. It will appear that it can be subjected to a peculiar transforma-
tion without losing its character as a solution.

This transformation consists in multiplying the imputations (37:2) of

37.3.2. by a common (positive) numerical factor z. In this way the follow-
ing set of imputations obtains:

N = °> ~ z )

(38:1) ' (2 2

7" = (0, 0, 0, 0}

and the imputations which originate from
these by permuting the players 1,2, 3, 4.

In order that these be imputations, all their components must be ^ — 1
(i.e. the common value of the v((z)). As z > 0 this means only that
— z ^ — 1, i.e. we must have

(38:2)

0 < 2 ^ 1.

For z — 1 our (38:1) coincides with (37:2) of 37.3.2. It would not seem
likely a 'priori that (38:1) should be a solution for the same game for any
other z of (38:2). And yet a simple discussion shows that it is a solution
if and only if z > $ — i.e. when (38 :2) is replaced by

(38:3)

< 2 ^ 1 .

The importance of this family of solutions is further increased by the fact
that it can be extended to a certain three-dimensional piece surrounding
the center of the cube Q. We shall give the necessary discussion in full,
because it offers an opportunity to demonstrate a technique that may be of
wider applicability in these investigations.

The interpretation of these results will be attempted afterwards.

38.1.2. We begin by observing that consideration of the set V defined
by the above (38:1) for the game described by (37:1) in 37.1.2. (i.e. the
center of Q), could be replaced by consideration of the original set V of
(37:2) in 37.3.2. in another game. Indeed, our (38:1) was obtained from
(37:2) by multiplying by z. Instead of this we could keep (37:2) and
multiply the characteristic function (37:1) by 1/z; this would destroy the
normalization y = 1 which was necessary for the geometrical representation
by Q (cf. 34.2.2.) — but we propose to accept that.

What we are now undertaking can be formulated therefore as follows:

So far we have started with a given game, and have looked for solutions.
Now we propose to reverse this process, starting with a solution and looking

322

ZERO-SUM FOUR-PERSON GAMES

for the game. Precisely: we start with a given set of imputations V* and
ask for which characteristic function v(S) (i.e. games) this V is a solution. 1

Multiplication of the v(S) of (37:1) in 37.1.2. by a common factor means
that we still demand

(38:4) v(S) = 0 when S is a two-element set,

but beyond this only the reduced character of the game (cf. 27.1.4.), i.e.

(38:5) v((l)) = v((2)) « v((3)) - v((4)).

Indeed, this joint value of (38:5) is — l/z and therefore (38:4), (38:5) and
(25:3:a) (25:3:b) in 25.3.1. yield that this v(S) is just (37:1) multiplied by
l/z. Our assertion (38:3) above means that the V of (37:2) in 37.3.2. is a
solution for (38:4), (38:5) if and only if the joint value of (38:5) (i.e. —l/z)
is ^ — 1 and > — f.

38 . 1 . 3 . Now we shall go one step further and drop the requirement of
reduction, i.e. (38:5). So we demand of v(S) only (38:4), restricting its
values for two-element sets S. We restate the final form of our question:

(38 :A) Consider all zero-sum four-person games where

(38:6) v(S) = 0 for all two-element sets S.

For which among these is the set V of (37 :2) in 37.3.2. a solution?

It will be noted that since we have dropped the requirements of normal-
ization and reduction of v(£) all connections with the geometrical represen-
tation in Q are severed. A special manipulation will be necessary therefore
at the end, in order to put the results which we shall obtain back into the
framework of Q.

38.2. Exact Discussion

38 . 2 . 1 . The unknowns of the problem (38: A) are clearly the values

(38:7) v((l) > ) = -y h v((2)) = - y 2 , v((3)) = -t/ 8 , v((4)) = — 1/ 4 .

We propose to determine what restrictions the condition in (38 :A) actually
places on these numbers y i, y 2 , t/ 8 , 2 / 4 .

1 This reversed procedure is quite characteristic of the elasticity of the mathematical
method — for the kind and degree of freedom which exists there. Although initially it
deflects the inquiry into a direction which must be considered unnatural from any but
the strictest mathematical point of view, it is nevertheless effective; by an appropriate
technical manipulation it finally discloses solutions which have not been found in any
other way.

After our previous examples where the guidance came from heuristic considerations,
it is quite instructive to study this case where no heuristic help is relied on and solutions
are found by a purely mathematical trick, — the reversal referred to above.

For the reader who might be dissatisfied with the use of such devices (i.e. exclusively
technical and non-conceptual ones), we submit that they are freely and legitimately used
in mathematical analysis.

We have repeatedly found the heuristic procedure easier to handle than the rigorous
one. The present case offers an example of the oDnosite.

NEIGHBORHOOD OF THE CENTER

323

This game is no longer symmetric. 1 Hence the permutations of the
players 1,2, 3, 4 are now legitimate only if accompanied by the corresponding
permutations of y h y 2 , y 2) y 4 . 2

To begin with, the smallest component with which a given player k is
ever associated in the vectors of (37 :2) in 37.3.2., is — 1. Hence the vectors
will be imputations if and only if — 1 v((fc)) i.e.

(38:8) y k ^ 1 for k = 1,2, 3, 4.

The character of V as a set of imputations is thus established; let us
now see whether it is a solution. This investigation is similar to the proof
given in 36.2.3-5.

38 . 2 . 2 . The observations (36:5), (36:6), of 36.2.3., apply again. A two-

element set S = (i ) j) is effective for a = [a h a 2} a 8 , a 4 } when a, + a, ^ 0

(cf. (38:A)). Hence we have for the a ', a" of (37:2): In a " every two-

element set >S is effective. In a ' : No two-element set S which does not con-
tain the player 4 is effective, while those which do contain him, S = (1,4),

(2,4), (3,4) clearly are. However, if we consider S = (1,4), we may discard

the two others; S = (2,4) arises from it by interchanging 1 and 2, which
— >

does not affect a '; 3 S = (3,4) is actually inferior to it after 1 and 3 are
interchanged, since i ^ 0. 4
Summing up:

(38 :B) Among the two-element sets S, those given below are cer-

tainly necessary, and all others are certainly unnecessary:

(1,4) for a', 5 all for a".

Concerning three-element sets: Owing to the above we may exclude by

— y — >

(36:6) all three-element sets for a ", and for a ' those which contain (1,4) or

(2,4). 6 This leaves only S = (1,2,3) for a '.

Summing up:

(38 :C) Among the three-element sets S , the one given below is

certainly necessary, and all others are certainly unnecessary:

(1,2,3) for

1 Unless yi - y* - y% « t/ 4 .

* But there is nothing objectionable in such uses of the permutations of 1,2, 3, 4 as we
made in the formulation of (37:2) in 37.3.2.

3 This permutation and similar ones later are clearly legitimate devices in spite of
footnote 1 above. Observe footnote 1 on p. 309 and footnote 2 above.

4 As « — 1 we could discard all these sets, including >S — (1,4), when v((4)) — — 1;
i.e. when 2/4 ** 1, which is a possibility. But we are under no obligation to do this. We
prefer not to do it, in order to be able to treat 2/4 - 1 and 2/4 > 1 together.

6 And all permutations of 1,2, 3, 4; these modify a ' too.

6 The latter obtains from the former by interchanging 1 and 2, which does not affect a # .

324

ZERO-SUM FOUR-PERSON GAMES

We leave to the reader the verification of (30:5 :a) in 30.1.1., i.e. that no
— > — >

a 'of V dominates any 0 of V. (Cf. the corresponding part of the proof in
36.2.4. Actually the proof of (30:5:b), which follows, also contains the
necessary steps.)

38 . 2 . 3 . We next verify (30:5:b) in 30.1.1., i.e. that an imputation 0
which is undominated by the elements of V must belong to V.

Consider a 0 undominated by the elements of V. If any two of 0i, 0 2 ,
08, 04 were < 0, we could make these (by permuting 1,2, 3, 4) 0i, 0 2 < 0.

This gives a " H 0 with S = (1,2) of (38 :B). Hence at most one of 0i, 0 2 ,

08, 04 is < 0. If none is < 0, then all are ^ 0. So each component of 0

is ^ the corresponding component of a", and since both are imputations

(cf. footnote 2 on p. 311), it follows that they coincide, 0 = a and so
it is in V.

Hence precisely one of 0i, 0 2 , 0 3 , 04 is < 0. By permuting 1,2, 3, 4 we
can make it 0 4 .

If any two of 0i, 0 2 , 0 3 were < -y, we could make these (by permuting
1,2,3) 0i, 0 2 < £. Besides, 0 4 < 0. So the interchange of 3 and 4 gives

a ' H 0 with S = (1,2,3) of (38:C). Hence at most one of 0i, 0 2 , 0 3 is <

If noneis < i, then0i, 02 , 0 3 ^ i. Hence 0 4 ^ But 0 4 ^ v((4)) = — 2 / 4 ,

so this necessitates — 2/4 ^ — f, i.e. 2/4 ^ f. Thus we need 2/4 < $ to
exclude this possibility, and as we are permuting freely 1,2, 3, 4, we even need

(38:9) Vk <i for k = 1,2, 3, 4.

If this condition is satisfied, then we can conclude that precisely one of
0i, 0 2 , 0 3 is < £. By permuting 1,2,3, we can make it 0 3 .

So 0i, 02 = "ff, 03 = 0. If 04 S — l, 1 then each component of 0 is ^ the

— >

corresponding component of a ', and since both are imputations (cf. foot-
note 2 on p. 311) it follows that they coincide: 0 = a ' and so it is in V.
Hence 0 4 < — 1. Also 0 3 < So interchange of 1 and 3 gives

a'hV with 8 = (1,4) of (38 :B).

This, at last, is a contradiction, and thereby completes the verification
of (30:5 :b) in 30.1.1.

The condition (38:9), which we needed for this proof, is really necessary:
it is easy to verify that

0 '=

1 If v((4)) » —1, i.e. if y A * 1, then this is certainly the case; but we do not wish
to assume it. (Cf. footnote 4 on p. 323.)

NEIGHBORHOOD OF THE CENTER

325

is undominated by our V, and the only way to prevent it from being an
imputation is to have — 1 < v((4)) = — y 4 , i.e. 2/4 < f. 1 * * Permuting 1,2, 3, 4
then gives (38 :9).

Thus we need precisely (38:8) and (38:9). Summing up:

(38 :D) The V of (37:2) in 37.3.2. is a solution for a game of (38 :A)

(with (38:6), (38:7) there) if and only if

(38:10) 1 ^ y k < f for k = 1,2, 3, 4.

38.2.4. Let us now reintroduce the normalization and reduction which
we abandoned temporarily, but which are necessary in order to refer these
results to Q, as pointed out immediately after (38 :A).

The reduction formulae of 27.1.4. show that the share of the player k
must be altered by the amount where

«2 = -v((k)) + i{v((l)) + v((2)) + v((3)) + v((4))}

= Vk — 4(2/1 + 2/2 + 2/8 + 2/4)

and

7 = - i{v((l)) + v((2)) + v((3)) + v((4))}

= i(yi + 2/2 + + 2/4).

For a two-element set S = (i, j ), v(S) is increased from its original value 0 to

a? + = Vi + yi - 4 ( 2/1 + 2/2 + 2/3 + 2/4)

= b(y% + yi - yk - yi)

(fc, l are the two players other than i } j).

The above 7 is clearly ^ 1 > 0 (by (38:10)), hence the game is essential.
The normalization is now carried out by dividing the characteristic function,
as well as every player’s share, by 7. Thus for S = (i 1 j), v(S) is now
modified further to

a < + = o yi + 2// - yk - yi

7 2/1 + 2/3 + 2/3 + 2/4’

This then is the normalized and reduced form of the characteristic
function, as used in 34.2.1. for the representation by Q. (34:2) id. gives,
together with the above expression, the formulae

1 Observe that the failure of V to dominate this 0 ' could not be corrected by adding

0 ' to V (when ^ |). Indeed, 0' dominates a" — |0, 0,0,0) with S — (1,2,3), so

it would be necessary to remove a " from V, thereby creating new undominated
imputations, etc.

If t/i “ yt *■ 2/* “ J /4 ■ I, then a change of unit by J brings our game back to the

— > — ►

form (37:1) of 37.1.2., and it carries the above 0 ' into the a IV — (1,1,1, — 1 } of (37:3)
in 37.4.1. Thus further attempts to make our V over into a solution would probably
transform it gradually into (37:3), (37:4) in 37.4.1-2. This is noteworthy, since we
started with (37:2) in 37.3.2.

These connections between the two solutions (37:2) and (37:3), (37:4) should be
investigated further.

326

ZERO-SUM FOUR-PERSON GAMES

y\ — y% — Vi + 2/4
yi + 2/2 + 2/3 + yt
— l/i + 2/2 — + y<

y\ + ^2 + Vs + 2/4
- 2/1 - 2/2 + 2/8 + 2/4

2/i + 2/2 + 2/3 + 2/4
for the coordinates xi, xj, xs in Q.

38 . 2 . 6 . Thus (38:10) and (38:11) together define the part of Q in which
these solutions — i.e. the solution (37:2) in 37.3.2., transformed as indicated
above — can be used. This definition is exhaustive, but implicit. Let us
make it explicit. I.e., given a point of Q with the coordinates xi, X 2 , xs,
let us decide whether (38:10) and (38:11) can then be satisfied together (by
appropriate y h y iy y it y A ).

We put for the hypothetical y h y 2 , y A

(38:12) 2/i + 2/2 + 2/s + 2/4 = ~

z

with z indefinite. Then the equations (38:11) become

/ , 4x,

I yi - yi - yt + y* = — >

(38:12*)

- yi + y* - y» + yt = —>

Z

-2/1 - 2/2 + y» + 2/4 = “

(38:12) and (38:12*) can be solved with respect to y i, y t , y 3 , y <:

(38:13)

1 + Xi — Xi — Xi 1 — Xi + Xi — X)

2/i = ' yi = >

z z

l — Xi — Xi + Xi

1 + Xi + Xi + x»

Now (38:11) is satisfied, and we must use our freedom in choosing z to
satisfy (38:10).

Let w be the greatest and v the smallest of the four numbers

(38:14)

Ml = 1 + Xi — Xi — Xi, Ut = 1 — Xi + Xt — Xj,

M* = 1 — Xi — Xi + X S , U t — 1 + Xi + X* + X t .

These are known quantities, since x t , x t) x$ are assumed to be given.

Now (38:10) clearly means that 1 g v/z and that w/z < f , i.e. it means
that

(38:15) fu> < z 2 a v.

Obviously this condition can be fulfilled (for z) if and only if
138:16) fw < v.

NEIGHBORHOOD OF THE CENTER 327

And if (38:16) is satisfied, then condition (38:15) allows infinitely many
values — an entire interval — for z.

38.2.6. Before we draw any conclusions from (38:15), (38:16), we give
the explicit formulae which express what has become of the solution (37 :2)

of 37.3.2. owing to our transformations. We must take the imputations
— * — 1 1

a ', a ", loc. cit., add the amount a* to the component k (i.e. to the player k ’ s
share), and divide this by y.

These manipulations transform the possible values of the component k
— which are i, 0, —1 in (37:2) — as follows. We consider first k = 1, and
use the above expressions for a fc and y as well as (38:13). Then:

x* - Xi,

Xty

X* ~ X|.

For the other k — 2,3,4 these expressions are changed only in so far that their
Xi — X* — xt is replaced by — Xi + x 2 — x 8 , — Xi — x% + x 8 , Xi + x* + x 8 ,
respectively. 1

Summing up (and recalling (38:14)):

(38 :E) The component k is transformed as follows:

i goes into z/2 + u* — 1,

0 goes into Uk — 1,

— 1 goes into — z + u k — 1,
with the u h u 2 , u 3) u 4 of (38:14).

We leave it to the reader to restate (37 :2) with the modification (38 :E),
paying due attention to carrying out correctly the permutations 1,2, 3, 4
which are required there.

It will be noted that for the center — i.e. Xi = x 2 = x 3 = 0 — (38 :E)
reproduces the formulae (38:1) of 38.1.1., as it should.

38.2.7. We now return to the discussion of (38:15), (38:16).

Condition (38:16) expresses that the four numbers u h u% y u 3} u 4 of (38:14)
are not too far apart — that their minimum is more than f of their maximum
— i.e. that on a relative scale their sizes vary by less than 2:3.

This is certainly true at the center, where X\ = x% = x% = 0; there
ui, u% y u 3) uk are all =1. Hence in this case v = w = 1, and (38:15)

1 This is immediate, owing to the form of the equations (38:13), and equally by con-
sidering the influence of the permutations of the players 1,2, 3, 4 on the coordinates
z\ t xty xj as described in 34.3.2.

_ g0 e 8 into i + ai = ? + 4 yi - ( yi + y t + y> + Vi )
2 y 2/1 + yt + y» + y<

1

0 goes into — — ~ +

° y 7/. 4- i/«

= 2 + Xl ~

vi + v± + vA = r „- x ,-

2/1 + 2/S + y» + 2/4

V flLVVO iUVV . . . •</ I

y 2 /i + 2/2 + y* + 2/4

goes i„ to r!±2! - -1 -My, i- ». + y. + ».)
y 2/1 + 2/2 + 2/» + 2/4

328

ZERO-SUM FOUR-PERSON GAMES

becomes $ < z g 1 proving the assertions made earlier in this discussion
(cf. (38:3) in 38.1.1.).

Denote the part of Q in which (38:16) is true by Z. Then even a
sufficiently small neighborhood of the center belongs to Z. 1 So Z is a
three-dimensional piece in the interior of Q, containing the center in its own
interior.

We can also express the relationship of Z to the diagonals of Q, say to
/-Center- V7JJ. Z contains the following parts of that diagonal. (Use
Figure 64): on one side precisely C, on the other a little less than half of
J?.* We add that these solutions are different from the family of solu-
tions valid in (36 :B) and (36 :C) which were referred to in 36.3.

38.S. Interpretation of The Solutions

38 . 3 . 1 . The family of solutions which we have thus determined possesses
several remarkable features.

We note first that for every game for which this family is a solution
at all (i.e. in every point of Z) it gives infinitely many solutions.* And all
we said in 37.5.1. applies again: these solutions are finite sets of imputations 4
and possess the full symmetry of the game. 6 Thus there is no “discrimina-
tion” in any one of these solutions. Nor can the differences in “organi-
zational principles,” which we discussed loc. cit., be ascribed to them. There
is nevertheless a simple “organizational principle” that can be stated in a
qualitative verbal form, to distinguish these solutions. We proceed to
formulate it.

38 . 3 . 2 . Consider (38 :E), which expresses the changes to which (37:2)

in 37.3.2. is to be subjected. It is clear that the worst possible outcome
for the player k in this solution is the last expression (since this corresponds
to — 1), — i.e. — z + u k — 1. This expression is > or = — 1, according to
whether z is < or = Now u h u 2y u 2 , U\ are the four numbers of (38:14),

the smallest of which is v. By (38:15) z ^ v, i.e. always — z + w* — 1 ^ — 1,

and = occurs only for the greatest possible value of z, z = v } — and then
only for those k for which u* attains its minimum, v.

1 If Xi, xj, x% differ from 0 by < A then each of the four numbers u lf ut, u s , u 4 of (38:14)
is < 1 + A — I and > 1 - A * i; hence on a relative size they vary by < I : | — f.
So we are still in Z. In other words: Z contains a cube with the same center as Q y but
with A of Q’s (linear) size.

Actually Z is somewhat bigger than this, its .volume is about jAo of the volume of Q.

* On that diagonal x\ ** x» ** x # , so the u i, u*, u 4 are: (three times) 1 — X\ and

1 *f Zx\. So for xi 0, v — 1 — xi, w — 1 -f 3xi, hence (38:16) becomes x\ < J.

And for X\ 0, v — 1 -f 3xi, w — 1 — x it hence (38:16) becomes > — A. So the

intersection is this:

0 ^ Xi < i (this is precisely C)

0 ^ Xi > — A (B is 0 ^ X\ > — 1).

* The solution which we found contained four parameters: pi, pj, p Jf p 4 while the games
for which they are valid had only three parameters: x if x it x$.

4 Each one has 13 elements, like (37:2) in 37.3.2.

* In the center x x — x% — x, — 0 we have pi — pi — pi — p« (cf. (38:13)), i.e. sym-
metry in 1,2, 3, 4. On the diagonal Xj » x* ■ x% we have pi *■ pi — p* (cf. (38:13)), i.e.
symmetry in 1,2,3.

NEIGHBORHOOD OF THE CENTER

329

We restate this:

(38 :F) In this family of solutions, even as the worst possible outcome,

a player k faces, in general, something that is definitely better
than what he could get for himself alone, — i.e. v((fc)) = — 1.
This advantage disappears only when z has its greatest possible
value, z = v, and then only for those k for which the correspond-
ing number iii, w 2 , u 8 , in (38:14) attains the minimum in
(38:14).

In other words: In these solutions a defeated player is in general not
“exploited” completely, not reduced to the lowest possible level — the level
which he could maintain even alone, i.e. v((fe)) = — 1. We observed
before such restraint on the part of a victorious coalition, in the “milder”
kind of “discriminatory” solutions of the three-person game discussed in
33.1. (i.e. when c > — 1, cf. the end of 33.1.2.). But there only one player
could be the object of this restraint in any one solution, and this phenomenon
went with his exclusion from the competition for coalitions. Now there is
no discrimination or segregation; instead this restraint applies to all players
in general, and in the center of Q ((38:1) in 38.1.1., with z < 1) the solution
is even symmetric! 1

38 . 3 . 3 . Even when z assumes its maximum value v } in general only one
player will lose this advantage, since in general the four numbers u i, u 8 ,
Ui of (38:14) are different from each other and only one is equal to their
minimum v. All four players will lose it simultaneously only if U\, w 2 , u%, u A
are all equal to their minimum v — i.e. to each other — and one look at
(38:14) suffices to show that this happens only when x\ = x\ = x% = 0,
i.e. at the center.

This phenomenon of not “exploiting” a defeated player completely is a
very important possible (but by no means necessary) feature of our solu-
tions, — i.e. of social organizations. It is likely to play a greater role in the
general theory also.

We conclude by stating that some of the solutions which we mentioned,
but failed to describe in 36.3.2., also possess this feature. These are the
solutions in C of Figure 64. But nevertheless they differ from the solutions
which we have considered here.

1 There is a quantitative difference of some significance as well. Both in our present
set-up (four-person game, center of Q) and in the one referred to (three-person game in
the sense of 33.1.), the best a player can do (in the solutions which we found) is i, and
the worst is —1.

The upper limit of what he may get in case of defeat, in those of our solutions where
he is not completely t[ ‘ exploited/* is now — J (i.e. — z with } < z £ 1) and it was J then

(i.e. c with -1 & c < |). So this zone now covers the fraction j _ (Z'ij — “ j “ g*

i.e. 221 % of the significant interval, while it then covered 100%.

CHAPTER VIII

SOME REMARKS CONCERNING n £ 5 PARTICIPANTS

39. The Number of Parameters in Various Classes of Games
39.1. The Situation for n = 3,4

39.1. We know that the essential games constitute our real problem
and that they may always be assumed in the reduced form and with 7 = 1.
In this representation there exists precisely one zero-sum three-person
game, while the zero-sum four-person games form a three-dimensional
manifold. 1 We have seen further that the (unique) zero-sum three-person
game is automatically symmetric, while the three-dimensional manifold
of all zero-sum four-person games contains precisely one symmetric game.

Let us express this by stating, for each one of the above varieties of
games, how many dimensions it possesses, — i.e. how many indefinite param-
eters must be assigned specific (numerical) values in order to characterize a
game of that class. This is best done in the form of a table, given in Figure
65 in a form extended to all n ^ 3. 2 Our statements above reappear in
the entries n = 3,4 of that table.

39.2. The Situation for All n ^ 3

39.2.1. We now complete the table by determining the number of
parameters of the zero-sum n-person game, both for the class of all these
games, and for that of the symmetric ones.

The characteristic function is an aggregate of as many numbers v(S)
as there are subsets S in I = (1, • • • , n), — i.e. of 2\ These numbers are
subject to the restrictions (25:3:a)-(25:3:c) of 25.3.1., and to those due to the
reduced character and the normalization 7 = 1, expressed by (27:5) in
27.2. Of these (25:3 :b) fixes v( — £) whenever v(S) is given, hence it
halves the number of parameters: 3 so we have 2 n ~ l instead of 2\ Next
(25:3:a) fixes one of the remaining v(S) : v(©) ; (27 :5) fixes n of the remaining
v(S):v((l)), • * • , v((n)); hence they reduce the number of parameters
by n + l. 4 So we have 2 n ~ l — n — 1 parameters. Finally (25:3:c) need
not be considered, since it contains only inequalities.

39.2.2. If the game is symmetric, then v(S) depends only on the number
of elements p of S: v(S) = v p , cf. 28.2.1. Thus it is an aggregate of as
many numbers v„ as there are p = 0, 1, • * • , n , — i.e. n + 1. These

1 Concerning the general remarks, cf. 27 . 1 . 4 . and 27 . 3 . 2 .; concerning the zero-sum
three-person game cf. 29 . 1 . 2 .; concerning the zero-sum four-person game cf. 34 . 2 . 1 .

* There are no essential zero-sum games for n - 1,2!

* S and — S are never the same set!

4 8 “ ©r (1), * * * , (n) differ from each other and from each other's complements.

330

NUMBER OF PARAMETERS

331

numbers are subject to the restrictions (28:11 :a)-(28:ll:c) of 28.2.1.; the
reduced character is automatic, and we demand also Vi = —y = — 1.
(28:11 :b) fixes v„_ p when v p is given; hence it halves the numbers of those
parameters for which n — p yt p. When n — p = p 1 — i.e. n = 2p, which hap-
pens only when n is even, and then p = n/2 — (28:11 :b) shows that this v.

must vanish. So we have

n + 1

7 %

parameters if n is odd and 5 if n is even,

instead of the original n + 1. Next (28:11 :a) fixes one of the remaining
v p : Vo; and Vi = — 7 = — 1 fixes another one of the remaining v p : v%;

hence they reduce the number of parameters by 2: 2 so we have — ^ — - — 2

ft

or ^ — 2 parameters. Finally (28:11 :c) need not be considered since it
contains only inequalities.

39.2.3. We collect all this information in the table of Figure 65. We also
state explicitly the values arising by specialization to n = 3, 4, 5, 6, 7, 8, —
the first two of which were referred to previously.

Number
of players

All games

Symmetric games

3

0*

0*

4

3

0*

5

10

1

6

25

1

7

56

2

8

119

2

n

2*-* 1 - n - 1

2 — 2 for n odd

n

— — 2 for n even

* Denotes the game is unique.

Figure 65 . — Essential games. (Reduced and 7 — 1 .)

The rapid increase of the entries in the left-hand column of Figure 65
may serve as another indication, if one be needed, how the complexity of a
game increases with the number of its participants. It seems noteworthy

1 Contrast this with footnote 3 on p. 330!

* p - 0, 1 differ from each other and from each other's n — p. (The latter only
because of n £ 3.)

332

REMARKS CONCERNING n £ 5 PARTICIPANTS

that there is an increase in the right-hand column too, i.e. for the symmetric
games, but a much slower one.

40. The Symmetric Five -person Game
40.1. Formalism of the Symmetric Five-person Game

40.1.1. We shall not attempt a direct attack on the zero-sum five-person
game. The systematic theory is not far enough advanced to allow it;
and for a descriptive and casuistic approach (as used for the zero-sum,
four-person game) the number of its parameters, 10, is rather forbidding.

It is possible however to examine the symmetric zero-sum five-person
games in the latter sense. The number of parameters, 1, is small but not
zero, and this is a qualitatively new phenomenon deserving consideration.
For n = 3,4 there existed only one symmetric game, so it is for n = 5
that it happens for the first time that the structure of the symmetric game
shows any variety.

40.1.2. The symmetric zero-sum five-person game is characterized by the
v p , p — 0,1, 2, 3, 4, 5 of 28.2.1., subject to the restrictions (28:11 :a)-(28:ll:c)
formulated there. (28:11 :a), (28:11 :b) state (with 7 = 1)

(40:1) v 0 = 0, Vi = — 1, v 4 = 1, v 6 = 0

and v 2 = — v 3 , i.e.

(40:2) v 2 = — tj, v 3 = 77

Now (28:11 :c) states v p + q ^ v p + v q for p + q ^ 5 and we can subject
p, q to the further restrictions of (28:12) id. Therefore p = 1, q = 1,2, 1
and so these two inequalities obtain (using (40:1), (40:2)):

p = 1, q = 1: —2 ^ —77; p = 1, q = 2: —1 — 77 ^ 77;

i.e.

(40:3) - i g 77 ^ 2.

Summing up:

(40 :A) The symmetric zero-sum five-person game is characterized

by one parameter 77 with the help of (40:1), (40:2). The domain
of variability of 77 is (40:3).

40.2. The Two Extreme Cases

40.2.1. It may be useful to give a direct picture of the symmetric games
described above. Let us first consider the two extremes of the interval
(40:3):

77 = 2, — 7.

1 This is easily verified by inspection of (28:12), or by using the inequalities of foot-
note 2 on p. 250. These give 1 £ p £ f, 1 £ q £ 2; hence as p, q are integers, p ** 1,

q - 1,2.

THE SYMMETRIC FIVE-PERSON GAME

333

Consider first 17 = 2: In this case v(S) = — 2 for every two-element
set S; i.e. every coalition of two players is defeated. 1 Thus a coalition of
three (being the set complementary to the former) is a winning coalition.
This tells the whole story : In the gradual crystallization of coalitions, the
point at which the passage from defeat to victory occurs is when the size
increases from two to three, and at this point the transition is 100 %. 2

Summing up:

(40 :B) 17 = 2 describes a game in which the only objective of all

players is to form coalitions of three players.

40 . 2 . 2 . Consider next rj = — In this case we argue as follows:

1 ( 4

when S has ) elements.

* ( 2

A coalition of four always wins. 3

Now the above formula shows that a coalition of two is doing just as
well, pro rata , as a coalition of four; hence it is reasonable to consider the
former just as much winning coalitions as the latter. If we take this
broader view of what constitutes winning, we may again affirm that the
whole story of the game has been told: In the formation of coalitions, the
point at which the passage from defeat to victory occurs is when the size
increases from one to two; at this point the transition is 100%. 4

Summing up:

(40 :C) rj = — i describes a game in which the only objective of all

players is to form coalitions of two players.

40 . 2 . 3 . On the basis of (40 :B) and (40 :C) it would be quite easy to guess
heuristically solutions for their respective games. This, as well as the
exact proof that those sets of imputations are really solutions, is easy; but
we shall not consider this matter further.

Before we pass to the consideration of the other rj of (40:3) let us remark
that (40 :B) and (40 :C) are obviously the simplest instances of a general

1 Cf. the discussion in 35.1.1., particularly footnote 4 on p. 296.

2 One player is just as much defeated as two, four are no more victorious than three.
Of course a coalition of three has no motive to take in a fourth partner; it seems (heuristi-
cally) plausible that if they do they will accept him only on the worst possible terms. But
nevertheless such a coalition of four wins if viewed as a unit, since the remaining isolated
player is defeated.

8 In any zero-sum n-person game any coalition of n — 1 wins, since an isolated player
is always defeated. Cf. loc. cit. above.

4 One player is defeated, two or four players are victorious. A coalition of three
players is a composite case deserving some attention: v(S) is — 1 for a three-element set
S, i.e. it obtains from the \ of a two-element set by addition of — 1. Thus a Coalition of
three is no better than a winning coalition of two (which it contains) plus the remaining
isolated and defeated player separately. This coalition is just a combination of win-
ning and a defeated group whose situation is entirely unaltered by this operation.

334

REMARKS CONCERNING n £ 5 PARTICIPANTS

method of defining games. This procedure (which is more general than that
of Chapter X, referred to in footnote 4 on p. 296) will be considered exhaus-
tively elsewhere (for asymmetric games also). It is subject to some restric-
tions of an arithmetical nature ; thus it is clear that there can be no (essential
symmetric zero-sum) n-person game in which every coalition of p is winning
if p is a divisor of n, since then n/ p such coalitions could be formed and every-
body would win with no loser left. On the other hand the same requirement
with p = n — 1 does not restrict the game at all (cf. footnote 3, p. 333).

40.3. Connection between the Symmetric Five-person Game and the 1,2,3-symmetric

Four-person Game

40 . 3 . 1 . Consider now the rj in the interior of (40:3). The situation is
somewhat similar to that discussed at the end of 35.3. We have some
heuristic insight into the conditions at the two ends of (40:3) (cf. above).
Any point ?y of (40:3) is somehow “surrounded” by these end-points. More
precisely, it is their center of gravity if appropriate weights are used. 1
The remarks made loc. cit. apply again: while this construction represents
all games of (40:3) as combinations of the extreme cases (40 :B), (40 :C), it
is nevertheless not justified to expect that the strategies of the former can
be obtained by any direct process from those of the latter. Our experiences
in the case of the zero-sum four-person game speak for themselves.

There is, however, another analogy with the four-person game which
gives some heuristic guidance. The number of parameters in our case is the
same as for those zero-sum four-person games which are symmetric with
respect to the players 1,2,3; we have now the parameter rj which runs over

(40:3) - i ^ v ^ 2,

while the games referred to had the parameter Xi which varies over
(40:4) “1 g ^ 1.*

This analogy between the (totally) symmetric five-person game and the
1,2,3-symmetric four-person game is so far entirely formal. There is,
however, a deeper significance behind it. To see this we proceed as follows:

40 . 3 . 2 . Consider a symmetric five-person game T with its ri in (40:3).
Let us now modify this game by combining the players 4 and 5 into one
person, i.e. one player 4'. Denote the new game by T'. It is important to
realize that T' is an entirely new game: we have not asserted that in T
players 4 and 5 will necessarily act together, form a coalition, etc., or that
there are any generally valid strategical considerations which would moti-
vate just this coalition. 8 We have forced 4 and 5 to combine; we did this
by modifying the rules of the game and thereby replacing T by T'.

1 The reader can easily carry out this composition in the sense of footnote 1 on p. 304,
relying on our equations (40:1), (40:2) in 40.1.2.

1 Cf. 35.3.2. In the representation in Q used there, X\ *= x% — x t .

* This ought to be contrasted with the discussion in 36.1.2., where a similar combina-
tion of two players was formed under such conditions that this merger seemed strategi-
cally justified.

THE SYMMETRIC FIVE-PERSON GAME

335

Now T is a symmetric five-person game, while T' is a 1,2,3-symmetric
four-person game. 1 Given the 17 of T we shall want to determine the
x\ of T' in order to see what correspondence of (40:3) and (40:4) this defines.
Afterwards we shall investigate whether there are not, in spite of what was
said above, some connections between the strategies — i.e. the solutions — of
T and T'.

The characteristic function v'(S) of T f is immediately expressible in
terms of the characteristic function v(S) of T. Indeed:

v'((l)) = v((l)) = -1,
v'((3)) - v((3)) = -1,
v'((l,2)) = v((l,2)) = -77,
v'((2,3)) = v((2,3)) - - 77 ,
v'((2,4')) = v((2,4,5)) = 77 ,
v'((l,2,3)) = v((l,2,3)) = 77 ,
v'((l,3,4')) = v((l,3,4,5)) = 1,

v'((2)) = v((2)) = -1,
v'((4')) = v((4,5)) = - 77 ;
v'((I,3)) - v((l,3)) = - 77 ,
v'((l,4')) = v((l,4,5)) = 77 ,
v'((3,4')) - v((3,4,5)) = v)
v'((l,2,4')) = v((l,2,4,5)) = 1,
v'((2,3,4')) - v((2,3,4,5)) = 1;

and of course

v'(©) = v'((l,2,3,4')) = 0.

While T was normalized and reduced, T' is neither; and we must bring
T' into that form since we want to compute its x h X2, £ 3 , i.e. refer it to the
Q of 34.2.2.

Let us therefore apply first the normalization formulae of 27.1.4. They
show that the share of the player k = 1,2, 3, 4' must be altered by the
amount a* where

and

Hence

at - —v'((k)) + i{v'((l)) + v'((2)) + v'((3)) + v'((4'))},
7 = - i{v'((l)) + v'((2)) + v'((3)) + v'((4'))} .

3 + 77

000 ^^ 0 3(1 77 )

= «2 = «3 = — > «4 1 »

7 =

This 7 is clearly ^

^ = g > 0 (by (40:3)); hence the game is

essential. The normalization is now carried out by dividing every player’s
share by 7.

Thus for a two-element set S = (i, j), v'(S) is replaced by

y"(S) = V '(£) + <*°i +

V ; 7

Consequently a simple computation yields

v"((l,2)) = v"((l,3)) = v"((2,3)) = - 2{ l V + v l ) >
v"((l,4')) = v"((2,4')) = v"((3,4')) =

1 The participants in r are players 1,2, 3, 4, 5, who all have the same role in the original
T. The participants in V are players 1,2,3 and the composite player (4,5) : 4'. Clearly
1,2,3 have still the same role, but 4' is different.

336

REMARKS CONCERNING n £ 5 PARTICIPANTS

This then is the normalized and reduced form of the characteristic
function, as used in 34.2. for the representation by Q. (34:2) in 34.2.1
gives, together with the above expression, the formula

- 1

xi = x 2 = x z = -k~t —

3 + rj

Taking Xi = x* = x 8 for granted, this relation can also be written as follows:
(40:5) (3 - *0(3 + v) = 10.

Now it is easy to verify that (40:5) maps the rj- domain (40:3) on the
Xi-domain (40:4). The mapping is obviously monotone. Its details are
shown in Figure 66 and in the adjoining table of corresponding values of
Xi and rj. The curve in this figure represents the relation (40:5) in the x h
17-plane. This curve is clearly (an arc of) a hyperbola.

40 . 3 . 3 . Our analysis of the 1,2,3-symmetric four-person game has
culminated in the result stated in 36.3.2.: The game, i.e. the diagonal
7-Center-V777 in Q which represents them, is divided into five classes
A-E, each of which is characterized by a certain qualitative type of
solutions. The positions of the zones A-E on the diagonal 7-Center-
VIII , i.e. the interval — 1 ^ xi 2a 1, are shown in Figure 64.

The present results suggest therefore the consideration of the cor-
responding classes of symmetric five-person games r in the hope that some
heuristic lead for the detection of their solutions may emerge from their
comparison with the 1,2,3-symmetric four-person games T, class by class.

Using the table in Figure 66 we obtain the zones A-E in — £ ^ 77 ^ 2,
which are the images of the zones A-E in — 1 2a X\ £ 1. The details
appear in Figure 67.

A detailed analysis of the symmetric five-person games can be carried
out on this basis. It discloses that the zones A, B do indeed play the role
which we expect, but that the zones C, D , E must be replaced by others,
C', D'. These zones A-D' in — i ^ rj ^ 2 and their inverse images A-D'
in — 1 2* X\ 2s 1 (again obtained with the help of the table of Figure 66)
are shown in Figure 68.

It is remarkable that the Xi-diagram of Figure 68 shows more symmetry
than that of Figure 67, although it is the latter which is significant for the
1,2,3-symmetric four-person games.

40 . 3 . 4 . The analysis of symmetric five-person games has also some
heuristic value beyond the immediate information it gives. Indeed, by
comparing the symmetric five-person game r and the 1,2,3-symmetric
four-person game T' which corresponds to it, and by studying the differences
between their solutions, one observes the strategic effects of the merger
of players 4 and 5 in one (composite) player 4'. To the extent to which the
solutions present no essential differences (which is the case in the zones A,
5, as indicated above) one may say that this merger did not affect the really

338 REMARKS CONCERNING n £ 5 PARTICIPANTS

significant strategic considerations. 1 On the other hand, when such differ-
ences arise (this happens in the remaining zones) we face the interesting
situation that even when 4 and 5 happen to cooperate in r, their joint
position is dislocated by the possibility of their separation. 2

Space forbids a fuller discussion based on the rigorous concept of solutions.

1 Of course one must expect, in solutions of r, arrangements where the players 4 and 5
are ultimately found in opposing coalitions. It is clear that this can have no parallel
in r'. All we mean by the absence of essential differences is that those imputations in a
solution of T which indicate a coalition of 4 and 5 should correspond to equivalent imputa-
tions in the solution of r'.

These ideas require further elaboration, which is possible, but it would lead too far
to undertake it now.

1 Already in 22.2., our first discussion of the three-person game disclosed that the
division of proceeds within a coalition is determined by the possibilities of each partner
in case of separation. But this situation which we now visualize is different. In our
present r it can happen that even the total share of player 4 plus player 5 is influenced
by this “ virtual ” fact.

A qualitative idea of such a possibility is best obtained by considering this: When a
preliminary coalition of 4 and 5 is bargaining with prospective further allies, their bar-
gaining position will be different if their coalition is known to be indissoluble (in T')
than when the opposite is known to be a possibility (in T).

CHAPTER IX

COMPOSITION AND DECOMPOSITION OF GAMES

41. Composition and Decomposition

41.1. Search for n-person Games for Which All Solutions Can Be Determined

41.1.1. The last two chapters will have conveyed a specific idea of the
rapidity with which the complexity of our problem increases as the number
n of participants goes to 4,5, • • • etc. In spite of their incompleteness,
those considerations tended to be so voluminous that it must seem com-
pletely hopeless to push this — casuistic — approach beyond five participants. 1
Besides, the fragmentary character of the results gained in this manner very
seriously limits their usefulness in informing us about the general possibilities
of the theory.

On the other hand, it is absolutely vital to get some insight into the
conditions which prevail for the greater values of n. Quite apart from the
fact that these are most important for the hoped for economic and socio-
logical applications, there is also this to consider : With every increase of
n, qualitatively new phenomena appeared. This was clear for each of
n = 2,3,4 (cf. 20.1.1., 20.2., 35.1.3., and also the remarks of footnote 2
on p. 221), and if we did not observe it for n = 5 this may be due to our
lack of detailed information about this case. It will develop later, (cf. the
end of 46.12.), that very important qualitative phenomena make their
first appearance for n = 6.

41.1.2. For these reasons it is imperative that we find some technique
for the attack on games with higher n. In the present state of things we
cannot hope for anything systematic or exhaustive. Consequently the
natural procedure is to find some special classes of games involving many
participants 2 that can be decisively dealt with. It is a general experience
in many parts of the exact and natural sciences that a complete under-
standing of suitable special cases — which are technically manageable, but
which embody the essential principles — has a good chance to be the pace-
maker for the advance of the systematic and exhaustive theory.

We will formulate and discuss two such families of special cases. They
can be viewed as extensive generalizations of two four-person games — so
that each one of these will be the prototype of one of the two families.
These two four-person games correspond to the 8 corners of the cube Q,
introduced in 34.2.2.: Indeed, we saw that those corners presented only

1 As was seen in Chapter VIII, it was already necessary for five participants to
restrict ourselves to the symmetric case.

* And in such a manner that each one plays an essential role!

339

340 COMPOSITION AND DECOMPOSITION OF GAMES

two strategically different types of games — the comers /, V, VI t VII ,
discussed in 35.1. and the comers //, III y JF, VIII, discussed in 35.2.
Thus the corners I and VIII of Q are the prototypes of those generaliza-
tions to which this chapter and the following one will be devoted.

41.2. The First Type. Composition and Decomposition

41 . 2 . 1 . We first consider the corner VIII of Q, discussed in 35.2. As
was brought out in 35.2.2. this game has the following conspicuous feature:
The four participants fall into two separate sets (one of three elements and
the other of one) which have no dealings with each other. I.e. the players
of each set may be considered as playing a separate game, strictly among
themselves and entirely unrelated to the other set.

The natural generalization of this is to a game T of n = A; + l par-
ticipants, with the following property: The participants fall into two sets of
k and l elements, respectively, which have no dealings with each other.
I.e. the players of each set may be considered playing a separate game, say
A and H respectively, strictly among themselves and entirely unrelated
to the other set. 1

We will describe this relationship between the games T, A, H by the
following nomenclature: Composition of A, H produces r, and conversely
T can be decomposed into the constituents A, H. 2

41 . 2 . 2 . Before we deal with the above verbal definitions in an exact
way, some qualitative remarks may be appropriate:

First, it should be noted that our procedure of composition and decompo-
sition is closely analogous to one which has been successfully applied in
many parts of modern mathematics. 3 As these matters are of a highly
technical mathematical nature, we will not say more about them here.
Suffice it to state that our present procedure was partly motivated by those
analogies. The exhaustive but not trivial results, which we shall obtain

1 In the original game of 35.2. the second set consisted of one isolated player, who was
also termed a “dummy.” This suggests an alternative generalization to the above one:
A game in which the participants fall into two sets such that those of the first set play a
game strictly among themselves etc., while those of the second set have no influence upon
the game, neither regarding their own fate, nor that of the others. (These are then the
“dummies.”)

This is, however, a special case of the generalization in the text. It is subsumed in
it by taking the game H of the second set as an inessential one, i.e. one which has a definite
value for each one of its participants that cannot be influenced by anybody. (Cf. 27.3.1.
and the end of 43.4.2. A player in an inessential game could conceivably deteriorate his
position by playing inappropriately. We ought to exclude this possibility for a
“dummy” — but this point is of little importance.)

The general discussion, which we are going to carry out (both games A and H essen-
tial) will actually disclose a phenomenon which does not arise in the special case to which
the corner VIII of 35.2. belongs — i.e. the case of “dummies” (H inessential). The
new phenomenon will be discussed in 46.7., 46.8., and the case of “dummies” — where
nothing new happens — in 46.9.

* It would seem natural to extend the concepts of composition and decomposition to
more than 2 constituents. This will be carried out in 43.2., 43.3.

3 Cf . G . Birkhoff & S . MacLane: A Survev of Modem Alsebra. New York. 1941.
Chapt. XIII.

COMPOSITION AND DECOMPOSITION 341

and also be able to use for further interpretations are a rather encouraging
symptom from a technical point of view.

41 . 2 . 3 . Second, the reader may feel that the operation of composition
is of an entirely formal and fictitious nature. Why should two games,
A and H, played by two distinct sets of players and having absolutely no
influence upon each other, be considered as one game T?

Our result will disclose that the complete separation of the games A
and H, as far as the rules are concerned, does not necessarily imply the
same for their solutions. I.e.: Although the two sets of players cannot
influence each other directly, nevertheless when they are regarded as one
set, — one society — there may be stable standards of behaviour which
establish correlations between them. 1 The significance of this circumstance
will be elaborated more fully when we reach it loc. cit.

41 . 2 . 4 . Besides, it should be noted that this procedure of composition
is quite customary in the natural sciences as well as in economic theory.
Thus it is perfectly legitimate to consider two separate mechanical systems —
situated, to take an extreme case, say one on Jupiter and one on Uranus —
as one. It is equally feasible to consider the internal economies of two
separate countries — the connections between which are disregarded — as one.
This is, of course, the preliminary step before introducing the interacting
forces between those systems. Thus we could choose in our first example
as those two systems the two planets Jupiter and Uranus themselves
(both in the gravitational field of the Sun), and then introduce as interaction
the gravitational forces which the planets exert on each other. In out
second example, the interaction enters with the consideration of inter-
national trade, international capital movements, migrations, etc.

We could equally use the decomposable game r as a stepping stone
to other games in its neighborhood, which, in their turn, permit no decompo-
sition. 2

In our present considerations, however, these latter modifications will
not be considered. Our interest is in the correlations introduced by the
solutions referred to at the beginning of this paragraph.

41.3. Exact Definitions

41 . 3 . 1 . Let us now proceed to the strictly mathematical description
of the composition and decomposition of games.

Let k players 1 forming the set J = (1', • • • , A:') play the

game A; and Z players 1", • • • , Z", forpaing the set K = (1", • • • , Z")
play the game H. We re-emphasize that A and H are disjoint sets of
players and 8 that the games A and H are without any influence upon each

1 There is sortie analogy between this and the phenomenon noted before (cf. 21.3.,
37.2.1.) that a symmetry of the game need not imply the same symmetry in all solutions.

1 Cf. 35.3.3., applied to the neighborhood of corner /., which according to 35.2. is a
decomposable game. The remark of footnote 2 on p. 303 on perturbations is also
pertinent.

3 If the same players 1, • • • , n are playing simultaneously two games, then an
entirely different situation prevails. That is the superposition of games referred to in

342 COMPOSITION AND DECOMPOSITION OF GAMES

other. Denote the characteristic functions of these two games by vaOS)
and v H (T) respectively, where S Qj and T Q K.

In forming the composite game r, it is convenient to use the same
symbols 1', • • • , 1", • • • , V for its n = k + l players. 1 They

form the set / = J u K = (1', - • 1", ••• , l").

Clearly every set R £ J permits a unique representation

(41:1) R = Su!T, SqJ, TqK;

the inverse of this formula being

(41:2) RnJ, T = R n K*

Denote the characteristic function of the game r by vr(R) with R £ J.
The intuitive fact that the games A and H combine without influencing
each other to T has this quantitative expression: The value in T of a coalition
R £ / obtains by addition of the value in A of its part S (s J) in J and of
the value in H of its part T (c K) in K. Expressed by a formula:

(41:3) Vr (R) = Va OS) + v H (T) where /?, S , T are linked by (41:1),

i.e. (41 :2). 8

41.3.2. The form (41:3) expressed the composite vr (R) by means of its
constituents va(£), v h (T). However, it also contains the answer to the
inverse problem: To express Va(£), v h (T) by v r (fl).

Indeed va(©) = v H (©) = 0. 4 Hence putting alternately T = ©
and S = © in (41:3) gives:

(41:4) vaOS) = vr(S) for SeJ,

(41:5) v H (T) = v r (T) for T c K. b

We are now in a position to express the fact of the decomposability of the
game T with respect to the two sets J and K. I.e.: the given game T
(among the elements of / = / u K) is such that it can be decomposed into
two suitable games A (among the elements J) and H (among the elements
of K ). As stated, this is an implicit property of r involving the existence
of the unknown A, H. But it will be expressed as an explicit property of T.

Indeed: If two such A, H exist, then they cannot be anything but those
described by (41 :4), (41 :5). Hence the property of r in question is, that the

27.6.2. and also in 35.3.4. Its influences on the strategy are much more complex and
scarcely describable by general rules, as was pointed out at the latter loc. cit.

1 Instead of the usual 1, • • • , n.

* These formulae (41:1), (41 :2) have an immediate verbal meaning. The reader may
find it profitable to formulate it.

* Of course, a rigorous deduction on the basis of 25.1.3. is feasible without difficulty.
All of 25.3.2. applies in this case.

* Note that the empty set © is a subset of both J and K ; since J and K are disjoint,
it is their only common subset.

* This is an instance of the technical usefulness of our treating the empty set © as a
coalition. Cf. footnote 2 on p. 241.

COMPOSITION AND DECOMPOSITION 343

A, H of (41:4), (41:5) fulfill (41:3). Substituting, therefore, (41:4), (41:5)
into (41 :3), using (41 :1) to express R in terms of S, T gives this:

(41 :6) v r (S uT) = v r (S) + v r (T) for SsJ, T S K.

Or, if we use (41 :2) (expressing S,,T in terms of R) in place of (41 :1)

(41 :7) v r (E) = v r (fl n J) + v r (R n K) for BcJ.

41 . 3 . 3 . In order to see the role of the equations (41:6), (41:7) in the
proper light, a detailed reconsideration of the basic principles upon which
they rest, is necessary. This will be done in sections 41.4.-42.5.2. which
follow. However, two remarks concerning the interpretation of these
equations can be made immediately.

First: (41:6) expresses that a coalition between a set S Q J and a set
T c K has no attraction — that while there may be motives for players
within J to combine with each other, and similarly for players within K 1
there are no forces acting across the boundaries of J and K.

Second: To those readers who are familiar with the mathematical theory
of measure, we make this further observation in continuation of that
made at the end of 27.4.3.: (41:7) is exactly CarathSodory’s definition of
measurability. This concept is quite fundamental for the theory of additive
measure and Carath6odory’s approach to it appears to be the technically
superior one to date. 1 Its emergence in the present context is a remarkable
fact which seems to deserve further study.

41.4. Analysis of Decomposability

41 . 4 . 1 . We obtained the criteria (41:6), (41:7) of T’ s decomposability
by substituting the v A (S), v H (T) obtained from (41:4), (41:5) into the
fundamental condition (41 :3). However, this deduction contains a lacuna:
We did not verify whether it is possible to find two games A, H which pro-
duce the v a (£), v h (T) formally defined by (41:4), (41:5).

There is no difficulty in formalizing these extra requirements. As we
know from 25.3.1. they mean that v A (S) and v H (T) fulfill the conditions
(25:3:a)-(25:3:c) eod. It must be understood that we assume the given
Vr (R) to originate from a game r, i.e. that vr (R) fulfills these conditions.
Hence the following question presents itself:

(41 :A) v r (R) fulfills (25:3:a)-(25:3:c) in 25.3.1. together with the

above (41:6), i.e. (41:7). Will then the v A 0S) and v H (T) of
(41:4), (41:5) also fulfill (25:3:a)-(25:3:c) in 25.3.1.? Or, if this
is not the case, which further postulate must be imposed upon
v r («)?

To decide this, we check (25:3:a)-(25:3:c) of 25.3.1. separately for va(S)
and v H (T). It is convenient to take them up in a different order.

41 . 4 . 2 . Ad (25:3:a): By virtue of (41:4), (41:5), this is the same state-
ment for va(S) and v H (T) as for vr (R).

1 Cf. C. Carothiodory: Vorlesungen liber Reelle Funktionen, Berlin, 1918, Chapt. V.

344 COMPOSITION AND DECOMPOSITION OF GAMES

Ad (25:3:c): By virtue of (41:4), (41:5), this carries over to v A (S) and
v h (T) from Vr(fi) — it amounts only to a restriction from the R Q I to
S QJ and T qK.

Before discussing the remaining (25:3:b), we insert a remark concerning
(25:4) of 25.4.1. Since this is a consequence of (25:3:a)-(25:3:c), it is
legitimate for us to draw conclusions from it — and it will be seen that this
anticipation simplifies the analysis of (25:3:b).

From here on we will have to use promiscuously complementary sets in
7, J, K. It is, therefore, necessary to avoid the notation — and to write
instead I — S, J — S y K — S, respectively.

Ad (25:4) : For v A OS) and v H (T) the role of the set 7 is taken over by the
sets J and K , respectively. Hence this condition becomes:

VA (J) = 0,

vrfUO = o.

Owing to (41:4), (41 :5), this means

(41:8) v r (J) = 0,

(41:9) v r (K) = 0.

Since K = 7 — J , therefore (25:3:b) (applied to v r 0S) for which it was
assumed to hold) gives

(41:10) Vr(t/) + Vr(7Q = 0.

Thus (41:8) and (41:9) imply each other by virtue of the identity (41:10).

In (41:8) or (41:9) we have actually a new condition, which does not
follow from (41:6) or (41:7).

Ad (25:3:b): We will derive its validity for v A 0S) and v H (T’) from the
assumed one for v r (R). By symmetry it suffices to consider VaOS).

The relation to be proven is

(41:11) vaOS) + Va(J - S) = 0.

By (41 :4) this means

(41:12) vr(S) + v r (7 - S) = 0.

Owing to (41:8), which we must require anyhow, this may be written
(41:13) VrOS) + v r (7 - S) « v r (J)

(Of course, S Q J .)

To prove (41:13), apply (25:3:b) for v r (R) to R = J - S and R = J.
For these sets 7 — R = S u K and 7 — R = K, respectively. So (41:13)
becomes

VrOS) - VrOSuTQ = -v r (K),
i.e.

VrOS uK) = vrOS) + v r (7C),
and this is the special case of (41:6) with T = K.

MODIFICATION OF THE THEORY 345

Thus we have filled in the lacuna mentioned at the beginning of this
paragraph and answered the questions of (41: A).

(41 :B) The further postulate which must be imposed upon vr (R)

is this: (41:8), i.e. (41:9).

All these put together answer the question of 41.3.2. concerning decom-
posability:

(41 :C) The game r is decomposable with respect to the sets J and K

(cf. 41.3.2.) if and only if it fulfills these conditions: (41:6), i.e.
(41:7) and (41:8), i.e. (41:9).

41.5. Desirability of a Modification

41.5.1. The two conditions which we proved equivalent to decompos-
ability in (41 :C) are of very different character. (41:6) (i.e. (41:7)) is the
really essential one, while (41 :8) (i.e. (41 :9)) expresses only a rather inciden-
tal circumstance. We will justify this rigorously below, but first a quali-
tative remark will be useful. The prototype of our concept of decomposition
was the game referred to at the beginning of 41.2.1.: the game represented
by the corner VIII of 35.2. Now this game fulfilled (41:6), but not
(41:8). (The former follows from (35:7) in 35.2.1., the latter from v(J) =
v((l,2,3)) = 1^0.) We nevertheless considered that game as decom-
posable (with J = (1,2,3), K = (4)) — how is it then possible, that it
violates the condition (41 :8) which we found to be necessary for the decom-
posability?

41.5.2. The answer is simple: For the above game the constituents A
(in J = (1,2,3)) and H (in K = (4)) do not completely satisfy (25:3:a)-
(25:3:c) in 25.3.1. To be precise, they do not fulfill the consequence
(25:4) in 25.4.1.: v A (J) = Vh(K) = 0 is not true (and it was from this
condition that we derived (41:8)). In other words: the constituents of T
are not zero-sum games. This point, of course, was perfectly clear in
35.2.2., where it received due consideration.

Consequently we must endeavor to get rid of the condition (41:8),
recognizing that this may force us to consider other than zero-sum games.

42. Modification of the Theory

42.1. No Complete Abandoning of the Zero-sum Condition

42.1. Complete abandonment of the zero-sum condition for our games 1
would mean that the functions 3C*(ri, • • • , r„) which characterized it
in the sense of 11.2.3. are entirely unrestricted. I.e. that the requirement

n

(42:1) X K *( ri - • • • , *•) - 0

fc -1

1 We again denote the players by 1, • • • , n.

346 COMPOSITION AND DECOMPOSITION OF GAMES

of 11.4. and 25.1.3. is dropped, with nothing else to take its place. This
would necessitate a revision of considerable significance, since the construc-
tion of the characteristic function in 25. depended upon (25:1), i.e. (42:1), and
would therefore have to be taken up de novo.

Ultimately this revision will become necessary (cf. Chapter XI) but not
yet at the present stage.

In order to get a precise idea of just what is necessary now, let us make
the auxiliary considerations contained in 42.2.1., 42.2.2. below.

42.2. Strategic Equivalence. Constant-sum Games

42 . 2 . 1 . Consider a zero-sum game r which may or may not fulfill
conditions (41:6) and (41:8). Pass from T to a strategically equivalent
game T' in the sense of 27.1.1., 27.1.2., with the a° lf • • • , aj described
there. It is evident, that (41 :6) for r is equivalent to the same for r'. 1

The situation is altogether different for (41:8). Passage from T to
T' changes the left hand side of (41:8) by ^ a?, hence the validity of

k in J

(41:8) in one case is by no means implied by that in the other. Indeed
this is true:

(42: A) For every T it is possible to choose a strategically equivalent

game T' so that the latter fulfills (41:8).

n

Proof : The assertion is 1 that we can choose «5, • • • , al with £ = 0

1

(this is (27:1) in 27.1.1.) so that

v(-/) + Z al = 0

k in J

Now this is obviously possible if J © or /, since then ^ a* can be

k in J

given any assigned value. For J = © or /, there is nothing to prove, as
then v(J) = 0 by (25:3:a) in 25.3.1. and (25:4) in 25.4.1.

This result can be interpreted as follows: If we refrain from considering
other than zero-sum games, 2 then condition (41:6) expresses that while
the game T may not be decomposable itself, it is strategically equivalent
to some decomposable game T'. 3

42 . 2 . 2 . The above rigorous result makes it clear where the weakness
of our present arrangement lies. Decomposability is an important strategic
property and it is therefore inconvenient that of two strategically equivalent
games one may be termed decomposable without the other. It is, therefore,

1 By (27:2) in 27.1.1. Observe that the Vr0$), v r '0$) of (42:A) are the v(£), v'(S) of
(27:2) loc. cit.

* I.e. we require this not only for r, but also for its constituents A, H.

* The treatment of the constituents in 35.2.2. amounts to exactly this, as an inspec-
tion of footnote 1 on p. 300 shows explicitly.

MODIFICATION OF THE THEORY 347

desirable to widen these concepts so that decomposability becomes an
invariant under strategic equivalence.

In other words: We want to modify our concept so that the transforma-
tion (27:2) of 27.1.1., which defines strategic equivalence, does not interfere
with the relationship between a decomposable game T and its constituents
A and H. This relationship is expressed by (41:3):

(42:2) vrGS u T) = v A (S) + v H (70 for SqJ, T c K.

Now if we use (27:2) with the same a° k for all three games T, A, H then (42:2)
is manifestly undisturbed. The only trouble is with the preliminary
condition (27:1). This states for r, A, H that

X at = 0, l a° = 0, 2 = °>

k in I k in J k in K

respectively — and while we now assume the first relation true, the two others
may fail.

Hence the natural way out is to discard (27:1) of 27.1.1. altogether.
I.e. to widen the domain of games, which we consider, by including all
those games which are strategically equivalent to zero-sum ones by virtue
of the transformation formula (27:2) alone — without demanding (27:1).

As was seen in 27.1.1. this amounts to replacing the functions

3C*(ri, * • * , r n )

of the latter by new functions

3C*( r i> t T ») ~ 3£k(j i, f T n ) -f- a*.

(The * ' ’ > a n are no longer subject to (27:1)). The systems of
functions 3C* On, • • • , r n ) which are obtained in this way from the system
of functions X fc (ri, • • • , r„) which fulfill (42:1) in 42.1. are easy to char-
acterize. The characteristic is (in place of (42:1) loc. cit.) the property

(42:3) £ 3Ci(r lf •",%)= s.>

k * 1

Summing up:

(42 :B) We are widening the domain of games which we consider, by

passing from the zero-sum games to the constant-sum games. 1 2
At the same time, we widen the concept of strategic equivalence,

1 8 is an arbitrary constant = 0. In the transformation (27:2) which produces this

<

game from a zero-sum one, there is obviously

n

a °k = S ‘

k - 1

* This gives a precise meaning to the statement at the beginning of 42.1. according to
which we are not yet prepared to consider all games unrestrictedly.

348 COMPOSITION AND DECOMPOSITION OF GAMES

introduced in 27.1.1., by defining it again by transformation
(27:2) loc. cit., but dropping the condition (27:1) eod.

42 . 2 . 3 . It is essential to recognize that our above generalizations do not
alter our main ideas on strategic equivalence. This is best done by con-
sidering the following two points.

First, we stated in 25.2.2. that we proposed to understand all quanti-
tative properties of a game by means of its characteristic function alone.
One must realize that the reasons for this are just as good in our present
domain of constant-sum games as in the original (and narrower) one of
zero-sum games. The reason is this:

(42 :C) Every constant-sum game is strategically equivalent to a

zero-sum game.

Proof: The transformation (27:2) obviously replaces the « of (42:3)

n

above by s + £ «2- Now it is possible to choose the a®, • • • , or® so

*-i

n

as to make this s + = 0, i.e. to carry the given constant-sum game

*-i

into a (strategically equivalent) zero-sum game.

Second, our new concept of strategic equivalence was only necessary
for the sake of the new (non-zero-sum) games that we introduced. For the
old (zero-sum) games it is equivalent to the old concept. In other words :
If two zero-sum games obtain from each other by means of the transforma-
tion (27:2) in 27.1.1., then (27:1) is automatically fulfilled. Indeed, this
was already observed in footnote 2 on p. 246.

42.3. The Characteristic Function in the New Theory

42 . 3 . 1 . Given a constant-sum game T' (with the 3C[(r h • • • , r n )
fulfilling (42:3)), we could introduce its characteristic function v'(£) by
repeating the definitions of 25. 1.3. 1 On the other hand, we may follow
the procedure suggested by the argumentation of 42.2.2., 42.2.3.: We can
obtain T f with the functions JC£(ri, • • • , r n ) from a zero-sum game F
with the functions 3 C*(ti, • • • , t„) as in 42.2.2., i.e. by

(42:4) OCKn, • • • , r.) ■ 3C*(r,, • • • , r n ) + *° k

with appropriate aj, • • • , a® (cf. footnote 1 on p. 246), and then define the
characteristic function v'(S) of F' by means of (27:2) in 27.1.1., i.e. by

(42:5) v'OS) - v(S) + £ a*°.

k in S

1 The whole arrangement of 25.1.3. can be repeated literally, although T' is no longer

zero-sum, with two exceptions. In (25:1) and (25:2) of 25.1.3. we must add s to the
extreme right hand term. (This is so, because we now have (42:3) in place of (42.1).)
This difference is entirely immaterial.

MODIFICATION OF THE THEORY

349

Now the two procedures are equivalent, i.e. the v'OS) of (42:4), (42:5)
coincides with the one obtained by the reapplication of 25.1.3. Indeed, an
inspection of the formulae of 25.1.3. shows immediately, that the substitu-
tion of (42:4) there produces the result (42 iS). 1 * 2

42 . 3 . 2 . v(£) is a characteristic function of a zero-sum game, if and only
if it fulfills the conditions (25:3:a)-(25:3:c) of 25.3.1., as was pointed out
there and in 26.2. (The proof was given in 25.3.3. and 26.1.) What do
these conditions become in the case of a constant-sum game?

In order to answer this question, let us remember, that (25:3:a)-(25:3:c)
loc. cit. imply (25:4) in 25.4.1. Hence, we can add (25:4) to them, and
modify (25:3:b) by adding v(/) to its right hand side (this is no change
owing to (25:4)). Thus the characterization of the v(S) of all zero-sum
games becomes this:

(42:6:a) v(©) - 0,

(42:6:b) v(S) + v(-S) = v(J),

(42:6:c) v(fl) + v(T) g v(S u T) if 8 n T - ©,
and

(42:6:d) v(7) = 0.

Now the v'(S) of all constant-sum games obtain from these \(S) by subject-
ing them to the transformation (42:5) of 42.3.1. How does this transforma-
tion affect (42:6:a)-(42:6:d)?

One verifies immediately, that (42:6:a)-(42:6:c) are entirely unaffected,
while (42:6:d) is completely obliterated. 3 So we see:

(42 :D) v(S) is the characteristic function of a constant-sum game

if and only if it satisfies the conditions (42:6:a)-(42:6:c).

(We write from now on v(S) for v'OS)).

As mentioned above, (42:6:d) is no longer valid. However, we have

(42:6:d*) v(7) = *.

Indeed, this is clear from (42:3), considering the procedure of 25.1.3. It
can also be deduced by comparing footnote 1 on p. 347 and footnote 3
above (our v(S) is the v'(S) there). Besides (42:6:d*) is intuitively clear:
A coalition of all players obtains the fixed sum $ of the game.

1 The verbal equivalent of this consideration is easily found.

1 Had we decided to define v'OS) by means of (42:2), (42:5) only, a question of ambi-
guity would have arisen. Indeed: A given constant-sum game r' can obviously be
obtained from many different zero-sum games r by (42:4), will then (42:5) always yield
the same v'OS)?

It would be easy to prove directly that this is the case. This is unnecessary, how-
ever, because we have shown that the v'OS) of (42:5) is always equal to that one of
25.1.3. — and that v'OS) is defined unambiguously, with the help of r' alone.

n

1 According to (42:5), the right hand side of (42:6:d) goes over into ^ a®, i.e. ^ aj,

tin I i — 1

and this sum is completely arbitrary.

350 COMPOSITION AND DECOMPOSITION OF GAMES

42.4. Imputations, Domination, Solutions in the New Theory

42 . 4 . 1 . From now on, we are considering characteristic functions of any
constant-sum game, i.e. functions v(S) subject to (42:6:a)-(42:6:c) only.

Our first task in this wider domain, is naturally that of extending to it
the concepts of imputations, dominations, and solutions as defined in 30.1.1.

Let us begin with the distributions or imputations. We can take over
from 30.1.1. their interpretation as vectors

a = {ai, • * * , a B ).

Of the conditions (30:1), (30:2) eod. we may conserve (30:1):

(42:7) a{ ^ v((»))

unchanged — the reasons referred to there 1 are just as valid now as then.
(30:2) eod., however, must be modified. The constant-sum of the game
being $ (cf. (42:3) and (42:6:d*) above), each imputation should distribute
this amount — i.e. it is natural to postulate

(42:8) £ «< = s.

i- i

By (42:6:d*) this is equivalent to

(42:8*) £ = v(I).*

»-l

The definitions of effectivity ) domination , solution we take over unchanged
from 30. 1.1. 3 the supporting arguments brought forward in the discussions
which led up to those definitions, appear to lose no strength by our present
generalization.

42 . 4 . 2 . These considerations receive their final corroborations by observ-
ing this:

(42 :E) For our new concept of strategic equivalence of constant-sum

games T, T', 4 there exists an isomorphism of their imputations,
i.e. a one-to-one mapping of those of r on those of T', which
leaves the concepts of 30. 1.1. 5 invariant.

This is an analogue of (31 :Q) in 31.3.3. and it can be demonstrated in
the same way. As there, we define the correspondence

(42:9)

1 ai < v((t)) would be unacceptable, cf. e.g. the beginning of 29.2.1.

* For the special case of a zero-sum game 8 « v(7) — 0 so (42:8), (42:8*) coincide — as
they must — with (30:2) loc. cit.

* I.e. (30:3); (30:4:a)-(30:4:c); (30:5:a), (30:5:b), or (30:5:c) loc. cit., respectively.

4 As defined at the end of 42.2.2., i.e. by (27:2) in 27.1.1., without (27:1) eod.

* As redefined in 42.4.1.

MODIFICATION OF THE THEORY

351

between the imputations a = {on, •••,<*») of T and the imputations

a' = {«;, • • • , a' } of r by

(42:10) a k = + cr°

where the a}, • • • , a® are those of (27:2) in 27.1.1.

Now the proof of (31 :Q) in 31.3.3. carries over almost literally. The one
difference is that (30:2) of 30.1.1. is replaced by our (42:8) but since (27:2)

n

in 27.1.1. gives v'(J) = v(/) + £ aj, this too takes care of itself. 1 The

t-i

reader who goes over 31.3. again, will see that everything else said there
applies equally to the present case.

42.6. Essentiality, Inessentiality, and Decomposability in the New Theory

42 . 5 . 1 . We know from (42 :C) in 42.2.3. that every constant-sum game is
strategically equivalent to a zero-sum game. Hence (42 :E) allows us to
carry over the general results of 31. from the zero-sum games to the constant-
sum ones always passing from the latter class to the former one by strategic
equivalence.

This forces us to define inessentiality for a constant-sum game by
strategic equivalence to an inessential zero-sum game. We may state
therefore:

(42 :F) A zero-sum game is inessential if and only if it is strategically

equivalent to the game with v(S) = 0. (Cf. 23.1.3. or (27 :C)
in 27.4.2.) By the above, the same is true for a constant-sum
game. (But we must use our new definitions of inessentiality
and of strategic equivalence.)

Essentiality is, of course, defined as negation of inessentiality.
Application of the transformation formula (42:5) of 42.3.1. to the
criteria of 27.4. shows, that there are only minor changes.

(27:8) in 27.4.1. must be replaced by

n

(42:11) 7 “ \ j v(/ ) “ 2 V(0 ’»)

since the right hand side of this formula is invariant under (42:5) and it
goes over into (27:8) loc. cit. for v(/) = 0 (i.e. the zero-sum case).

The substitution of (42:11) for (27:8) necessitates replacement of the 0
on the right hand side of both formulae in the criterion (27 :B) of 27.4.1. by

n

1 And this was the only point in the proof referred to, at which a® - 0 (i.e. (27:1)

»-i

in 27.1.1., which we no longer require) is used.

352 COMPOSITION AND DECOMPOSITION OF GAMES

v(J). The criteria (27 :C), (27 :D) of 27.4.2. are invariant under (42:5),
and hence unaffected.

42 . 5 . 2 . We can now return to the discussion of composition and decom-
position in 41.3.-41.4., in the wider domain of all constant-sum games.

All of 41.3. can be repeated literally.

When we come to 41.4.,, the question (41 :A) formulated there again
presents itself. In order to determine whether any postulates beyond
(41:6), i.e. (41:7) of 41.3.2. are now needed, we must investigate (42:6:a)-
(42:6:c) in 42.3.2., instead of (25:3:a)-(25:3:c) in 25.3.1. (for all three of
vr (ft), va(S), Vh(T)).

(42:6:a), (42:6:c) are immediately disposed of, exactly as (25:3:a),
(25:3:c) in 41.4. As to (42:6:b), the proof of (25 :3 :b) in 41.4. is essentially
applicable, but this time no extra condition arises (like (41:8) or (41:9)
loc. cit.). To simplify matters, we give this proof in full.

Ad (42:6 :b): We will derive its validity for Va(£) and v H (T) from the
assumed one for vr (R). By symmetry it suffices to consider VaGS).

The relation to be proven is

(42:12) VaGS) + VaG / - S) = v A (J).

By (41:4) this means

(42:12*) vrGS) + v r (J — S) = v r (J).

To prove (42:12*) apply (42:6:b) for vr (R) to R = J — S and R = J.
For these I — R = S u K and I — R = K, respectively. So (42:12*)
becomes

vrGS) + v r (J) - vr OS u K) = v r (J) - v r (X),
i.e.

vrGS u K) = v r OS) + v r (7Q,

and this is the special case of (41:6) with T = K.

Thus we have improved upon the result (41 :C) of 41.4. as follows:

(42 :G) In the domain of all constant-sum games the game r is

decomposable with respect to the sets J and K (cf. 41.3.2.) if
and only if it fulfills the condition (41:6), i.e. (41:7).

42 . 5 . 3 . Comparison of (41 :C) in 41.4. and of (42 :G) in 42.5.2. shows that
the passage from zero-sum to constant-sum games rids us of the unwanted
condition (41:8), i.e. (41:9) for decomposability.

Decomposability is now defined by (41:6), i.e. (41:7) alone, and it is
invariant under strategic equivalence — as it should be.

We also know that when a game r is decomposed into two (constituent)
games A and H (all of them constant-sum only!), we can make all these
games zero-sum by strategic equivalence. (Cf. (42 :C) in 42.2.3. for r,
and then (42 :A) in 42.2.1. et sequ. for A, H.)

THE DECOMPOSITION PARTITION

353

Thus we can always use one of the two domains of games — zero-sum
or constant-sum — whichever is more convenient for the problem just under
consideration.

In the remainder of this chapter we will continue to con-
sider constant-sum games, unless the opposite is explicitly stated.

43. The Decomposition Partition
43.1. Splitting Sets. Constituents

43.1. We defined the decomposability of a game T not per se y but with
respect to a decomposition of the set / of all players into two complementary
sets, J , K.

Therefore it is feasible to take this attitude: Consider the game T
as given, and the sets J ) K as variable. Since J determines K (indeed
K == / — J), it suffices to treat J as the variable. Then we have this
question:

Given a game T (with the set of players I) for which sets J £ I (and the
corresponding K = / — J) is r decomposable?

We call those J(Q I) for which this is the case the splitting sets of V.
The constituent game A which obtains in this decomposition (cf. 41.2.1.
and (41:4) of 41.3.2.) is the J -constituent of r. 1

A splitting set J is thus defined by (41:6), i.e. (41:7) in 41.3.2., where
K = / — J must be substituted.

The reader will note that this concept has a very simple intuitive mean-
ing: A splitting set is a self contained group of players, who neither influence,
nor are influenced by, the others as far as the rules of the game are concerned.

43.2. Properties of the System of All Splitting Sets

43.2.1. The totality of all splitting sets of a given game is characterized
by an aggregate of simple properties. Most of these have an intuitive
immediate meaning, which may make mathematical proof seem unnecessary.
We will nevertheless proceed systematically and give proofs, stating the
intuitive interpretations in footnotes. Throughout what follows we write
v(£) for Vr(S) (the characteristic function of T).

(43 : A) J is a splitting set if and only if its complement K = / — J

is one. 2

Proof : The decomposability of T involves J and K symmetrically.

(43 :B) © and I are splitting sets. 8

Proof : (41:6) or (41:7) with J = ©, K = I are obviously true, as
v(©) = 0.

1 By the same definition the game H (cf. 41.2.1. and (41:5) in 41.3.2.) is then the
^-constituent (K ■** / — J) of r.

* That a set of players is self-contained in the sense of 43.1., is clearly the same state-
ment, as that the complement is self-contained.

1 That these are self-contained is tautological.

354 COMPOSITION AND DECOMPOSITION OF GAMES

43 . 2 . 2 .

(43 :C) J' n J " and J' u «/" are splitting sets if J', J" are. 1

Proof: Ad J' u J": As «/', J" are splitting sets, we have (41:6) for J,
equal to J', I — J' and J", / — «/". We wish to prove it for J ' u J",
/—(«/' u «/"). Consider therefore two $ £ J' u J", T £ / — («/' u «/").
Let S' be the part of S in J', then S" = S — S' lies in the complement of
J' y and as S £ J' u J", S" also lies in J". So S = S' + S", S' £ J',
S" £ Now S' £ J', S" £ / - J' and (41 :6) for ■/', I — J' give

(43:1) v(S) = v(S') + v(S").

Next S" £ 7 — J' and T £ / - (J' u J") £ / - J' so S"uT£/-J'.
Also S' £ J'. Clearly S' u (S" u T) = S u T. Hence (41 :6) for J', I — J 1
also gives

(43:2) v(S u T) = v(S') + v(S" u 7 1 ).

Finally S"£J" and T £/-(/' u J") £/- J". Hence (41:6) for
J", / — J" gives

(43:3) v(S" ul 7 ) = v(S") + v(T).

Now substitute (43:3) into (43:2) and then contract the right hand side
by (43:1). This gives

v(S u T) = v(S) + v(F),
which is (41:6), as desired.

Ad J' n J Use (43: A) and the above result. As J', J" are splitting
sets, the same obtains successively for I — J', I — «/", (/ — J') u (/ — /")
which is clearly / — («/' n J") 2 , and J ' n «/" — the last one being the desired
expression.

43.3. Characterization of the System of All Splitting Sets. The Decomposition Partition

43 . 3 . 1 . It may be that there exist no other splitting sets than the trivial
ones ©, I (cf. (43 :B) above). In that case, we call the game T indecompos-
able.* Without studying this question any further, 4 we continue to
investigate the splitting sets of T.

1 The intersection J ' n J It may strike the reader as odd, that two self-contained
sets J J” should have a non-empty intersection at all. This is possible, however, as the
example J' — J" shows. The deeper reason is that a self-contained set may well be the
sum of smaller self-contained sets (proper subsets). (Cf. (43:H) in 43.3.) Our present
assertion is that if two self-contained sets J\ J " have a non-empty intersection /' 0
then this intersection is such a self-contained subset. In this form it will probably appear
plausible.

The sum J’ U J": That the sum of two self-contained sets will again be self-contained
stands to reason. This may be somewhat obscured when a non-empty intersection
J' n J " exists, but this case is really harmless as discussed above. The proof which
follows is actually primarily an exact account of the ramifications of just this case.

1 The complement of the intersection is the sum of the complements.

1 Actually most games are indecomposable; otherwise the criterion (42 :G) in 42.5.2.
requires the restrictive equations (41 :6), (41 :7) in 41.3.2.

4 Yet! Cf. footnote 3 and its references^

THE DECOMPOSITION PARTITION

355

(43 :D) Consider a splitting set / of r and the /-constituent A of r.

Then a /' £ / is a splitting set of A if and only if it is one of IV

Proof: Considering (41:4), J' is a splitting set of A by virtue of (41:6)
when

(43:4) v(S u T) = v(S) + v(T) for S £ /', T sj — J '.

(We write v(£) for v r (<S)). Again by (41 :6) /' is a splitting set of T when

(43:5) v(£ uT) = v(£) + v(T) for S£/', T&I-J'.

We must prove the equivalence of (43:4) and (43:5). As J £ J, so
(43:4) is clearly a special case of (43:5) — hence we need only prove that
(43:4) implies (43:5).

Assume, therefore, (43:4). We may use (41 :6) for T with J, K — I — J.
Consider two S £ /', T £ I — /'. Let T f be the part of T in J, then
T" = T - r lies in I - J. So T = V u T", 7 V £ J, T" £ / — / and
(41:6) for T with J, J — / give

(43:6) v(T) = v(T') + v(T").

Next S £ /' £ / and T' £ J so S u f £ J. Also T” £ / — /. Clearly
(S u T') u T n = >S u T. Hence (41 :6) for T with /, / — J also gives

(43:7) v(S uT) = v(S u T) + v(T").

Finally «£j' and T £/ - /' and T'£/, so T'sJ-J'. Hence
(43:4) gives

(43:8) v(Suf) = v(S) + v(T').

Now substitute (43:8) into (43:7) and then contract the right hand side
by (43:6). This gives precisely the desired (43:5).

43.3.2. (43 :D) makes it worth while to consider those splitting sets J,
for which / ^ ©, but no proper subset /' -A © of J is a splitting set. We
call such a set /, for obvious reasons, a minimal splitting set.

Consider our definitions of indecomposability and of minimality.
(43 :D) implies immediately:

(43 :E) The /-component A (of T) is indecomposable if and only if

/ is a minimal splitting set.

The minimal splitting sets form an arrangement with very simple
properties, and they determine the totality of all splitting sets. The
statement follows:

(43 :F) Any two different minimal splitting sets are disjunct.

(43 :G) The sum of all minimal splitting sets is I.

1 To be self-contained within a self-contained set, is the same thing as to be such in
the original (total) set. The statement may seem obvious; that it is not so, will appear
from the proof.

356 COMPOSITION AND DECOMPOSITION OF GAMES

(43 :H) By forming all sums of all possible aggregates of minimal

splitting sets, we obtain precisely the totality of all splitting
sets. 1

Proof: Ad (43 :F): Let J ', J" be two minimal splitting sets which are
not disjunct. Then J ' n J' V © is splitting by (43 :C), as it is £ J f and
£ J". So the minimality of J ' and J " implies that J f n J" is equal to
both J' and J". Hence J' = J".

Ad (43 :G) : It suffices to show that every & in I belongs to some minimal
splitting set.

There exist splitting sets which contain the player k (i.e. I ) ; let J be the
intersection of all of them. J is splitting by (43 :C). If J were not minimal,
then there would exist a splitting set J r ©, J, which is £j. Now
J" = J — J f = J n (7 — J') is also a splitting set by (43 :A), (43 :C), and
clearly also J" ©, J. Either J' or J" = J — J' must contain k — say
that J ' does. Then J ' is among the sets of which J is the intersection.
Hence J' 2. J. But as J' £ J and J ' ^ J , this is impossible.

Ad (43 :H): Every sum of minimal splitting sets is splitting by (43 :C),
so we need only prove the converse.

Let K be a splitting set. If J is minimal splitting, then J n K is splitting
by (43 :C), also J n K £ J — hence either J n K ~ Q or J n K = J. In
the first case J, K are disjunct, in the second J £K. So we see:

(43:1) Every minimal splitting set J is either disjunct with K or

si

Let K' be the sum of the former J, and K n the sum of the latter. K ' u K n
is the sum of all minimal splitting sets, hence by (43 :G)

(43:9) I'ul" = I.

By their origin K ' is disjunct with K y and K " is £ K. I.e.

(43:10) K' £ I — K, K" £ K.

Now (43:9), (43:10) together necessitate K " = K; hence If is a sum of a
suitable aggregate of minimal sets, as desired.

43 . 3 . 3 . (43 :F), (43 :G) make it clear that the minimal splitting sets
form a partition in the sense of 8.3.1., with the sum 7. We call this the
decomposition partition of T, and denote it by Ilr. Now (43 :H) can be
expressed as follows:

(43 :H*) A splitting set K£l is characterized by the following

property: The points of each element of Ilr go together as
far as K is concerned — t.e. each element of Ilr lies completely
inside or completely outside of K .

1 The intuitive meaning of these assertions should be quite clear. They characterize
the structure of the maximum possibilities of decomposition of r in a plausible way.

THE DECOMPOSITION PARTITION

357

Thus n r expresses how far the decomposition of T in I can be pushed,
without destroying those ties which the rules of T establish between players. 1
By virtue of (43 :E) the elements of II r are also characterized by the fact
that they decompose T into indecomposable constituents.

43.4. Properties of the Decomposition Partition

43 . 4 . 1 . The nature of the decomposition partition II r being established,
it is natural to study the effect of the fineness of this partition. We wish to
analyze only the two extreme possibilities: When Ilr is as fine as possible,
i.e. when it dissects / down to the one-element sets — and when Ilr is as coarse
as possible, i.e. when it does not dissect I at all. In other words: In the first
case Ilr is the system of all one-element sets (in I ) — in the second case Ilr
consists of I alone.

The meaning of these two extreme cases is easily established:

(43 :J) Ilr is the system of all one-element sets (in I) if and only if

the game is inessential.

Proof : It is clear from (43 :H) or (43 :H*) that the stated property of
n r is equivalent of saying that all sets /(£ I) are splitting. I.e. (by 43.1.)
that for any two complementary sets J and K(= I — J) the game T is
decomposable. This means that (41:6) holds in all those cases. This
implies, however, that the condition imposed by (41 :6) on S , T (i.e. S £ J,
T Q K) means merely that S , T are disjunct. Thus our statement becomes

vOS uT) = v(S) + v(T) for S n T = ©

Now this is precisely the condition of inessentiality by (27 :D) in 27.4.2.
(43 :K) n r consists of I if and only if the game T is indecomposable.

Proof: It is clear from (43:H) (or (43:H*)), that the stated property of
Ilr is equivalent to saying that ©, I are the only splitting sets. But this
is exactly the definition of indecomposability at the beginning of 43.3.

These result^ show that indecomposability and inessentiality are two
opposite extremes for a game. In particular, inessentiality means that the
decomposition of T, described at the end of 43.3., can be pushed thrdugh
to the individual players, without ever severing any tie that the rules of the
game T establish. 2 The reader should compare this statement with our
original definition of inessentiality in 27.3.1.

43 . 4 . 2 . The connection between inessentiality, decomposability, and
the number n of players is as follows:

n = 1: This case is scarcely of practical importance. Such a game is
clearly indecomposable, 8 and it is at the same time inessential by the first
remark in 27.5.2.

1 I.e. without impairing the self-containedness of the resulting sets.

2 I.e. that every player is self-contained in this game.

8 As / is a one-element set ©, I are its only subsets.

358 COMPOSITION AND DECOMPOSITION OF GAMES

It should then be noted that indecomposability and inessentiality are
by (43 :J), (43 :K) incompatible when n ^ 2, but not when n = 1.

n = 2: Such a game, too, is necessarily inessential by the first remark oi
27.5.2. Hence it is decomposable.

n ^ 3: For these games decomposability is an exceptional occurrence,
Indeed, decomposability implies (41:6) with some J ^ Q, /; hence K =
I — J 7* ©, /. So we can choose j in J t k in K. Then (41 :6) with S = ( j )
T = ( k ) gives

(43:11) v((j, * 0 ) = v((j)) + v((*)).

Now the only equations which the values of v(S) must satisfy, are (25:3:a),
(25:3:b) of 25.3.1. (if zero-sum games are considered) or (42:6:a), (42:6:b)
of 42.3.2. (43:11) is neither of these, since only the sets (j), ( k ), (j, k)

occur in (43:11) and these are none of the sets occurring in those equations —
i.e. © or / or complements — as n ^ 3. 1 Thus (43:11) is an extra conditior
which is not fulfilled in general.

By the above an indecomposable game cannot have n = 2 hence it has
n = 1 or w ^ 3. Combining this with (43 :E), we obtain the following
peculiar result:

(43 :L) Every element of the decomposition partition Ilr is either a

one-element set, or else it has n ^ 3 elements.

Note that the one-element sets in II r are the one-element splitting sets :
— i.e. they correspond to those players who are self-contained, separatee
from the remainder of the game (from the point of view of the strategy ol
coalitions). They are the “ dummies ” in the sense of 35.2.3. and footnote ]
on p. 340. Consequently, our result (43 :L) expresses this fact : Those players
who are not “dummies,” are grouped in indecomposable constituent games
of n ^ 3 players each.

This appears to be a general principle of social organization.

44. Decomposable Games. Further Extension of the Theory
44.1. Solutions of a (Decomposable) Game and Solutions of Its Constituents

44.1. We have completed the descriptive part of our study of composi
tion and decomposition. Let us now pass to the central part of the problem
The investigation of the solutions in a decomposable game.

Consider a game T which is decomposable for J and I — J = K, wit!
the J- and /^-constituents A and H. We use strategic equivalence, as
explained at the beginning of 42.5.3., to make all three games zero-sum.

Assume that the solutions for A as well as those for H are known; doei
this then determine the solution for r? In other words: How do tin
solutions for a decomposable game obtain from those for its constituents?

Now there exists a surmise in this respect which appears to be the primi
facie plausible one, and we proceed to formulate it.

1 For n — 2 it is otherwise; (j, k) = /, (J) and (k) are complements.

1 Such a splitting set is, of course, automatically minimal.

DECOMPOSABLE GAMES

359

44.2. Composition and Decomposition of Imputations and of Sets of Imputations

44.2.1. Let us use the notations of 41.3.1. But as we write v(S) for
Vr (S) this also replaces by (41:4), (41:5), VaOS), VhOS).

On the other hand, we must distinguish between imputations for
T, A, H. 1 In expressing this distinction, it is better to indicate the set of
players to whom an imputation refers, instead of the game in which they are
engaged. I.e. we will affix to them the symbols /, J> K rather than T, A, H.
In this sense we denote the imputations for I (i.e. T) by

(44:1) a i = jar, • • • , a^, ar, • * • , ar ),

and those for J , K (i.e. A, H) by

(44:2) (3j = \fi v , • • • ,/M,

(44:3) y k = |7r, • * • , 7r|.

If three such imputations are linked by the relationship

(44:4)

«»' = Pi'
«>" = 7,"

for

for

*" = 1',

3" « 1",

> ™ }
l"

y 1 y

then we say that a 7 obtains by composition for ft /, y K) that P Jy y K obtain

by decomposition from a i (for J, K ), and that P Jf y K are the (»/-, K - )
— >

constituents of a /.

Since we are now dealing with zero-sum games, all these imputations
must fulfill the conditions (30:1), (30:2) of 30.1.1. Now one verifies

immediately for a 7 , P Jf y K linked by (44:4).

— > — >

Ad (30:1) of 30.1.1. : The validity of this for 0 /, y * is clearly equivalent
to its validity for a /.

Ad (30:2) of 30.1.1.: For 0 j, y k this states (using (44:4))

X «<' = 0.

(44:5)

(44:6)

For a / it amounts to
(44:7)

t'-r

i"

£ «j" = o.
1"

k' l"

S X a >" ^

V j"- 1 "

1 It is now convenient to re-introduce the notations of 41.3.1. for the players.

360 COMPOSITION AND DECOMPOSITION OF GAMES

Thus its validity for P Jy y K implies the same for a /, while its validity

for a i does not imply the same for p Jy y K — indeed (44:7) does imply the
equivalence of (44:5) and (44:6), but it fails to imply the validity of either
one.

So we have:

(44: A) Any two imputations P j, y K can be composed to an «/,

— > — ► — >

while an imputation a / can be decomposed of two P j, y K if and

only if it fulfills (44:5), i.e. (44:6).

We call such an a / decomposable (for J, K ).

44 . 2 . 2 . This situation is similar to that- which prevails for the games
themselves: Composition is always possible, while decomposition is not.
Decomposability is again an exceptional occurrence. 1

It ought to be noted, finally, that the concept of composition of imputa-
tions has a simple intuitive meaning. It corresponds to the same operation
of “ viewing as one” two separate occurrences, which played the correspond-
ing role for games in 41.2.1., 41.2.3., 41.2.4. Decomposition of an a ,

(into P j } y K ) is possible if and only if the two self-contained sets of players

— >

J , K are given by the sets of imputations a / precisely their “just dues” —
which are zero. This is the meaning of the condition (44 :A) (i.e. of (44:5),
(44:6)).

44 . 2 . 3 . Consider a set Vj of imputations p j and a set Wx of imputations
— ^ — >

y K* Let U/ be the set of those imputations a / which obtain by composition
— ^ ^

of all p j in V/ with all y K in W*. We then say that U; obtains by compo-
sition from V/, Wxi that V/, W* obtain by decomposition from U/ (for
J, K) y and that V/, W k are the (J-, K-) constituents of U /.

Clearly the operation of composition can always be carried out, what-
ever V/, W k — whereas a given U/ need not allow decomposition (for J , K).
If U / can be decomposed, we call it decomposable (for J, K).

Note that this decomposability of U/ restricts it very strongly; it implies,

among other things that all elements a / of U/ must be decomposable (cf.
the interpretation at the end of 44.2.2.).

In order to interpret these concepts for the sets of imputations U/, Vj,
Wjc more thoroughly, it is convenient to restrict ourselves to solutions
of the games r, A, H.

1 There are great technical differences between the concepts of decomposability etc.,
for games and for imputations. Observe, however, the analogy between (41:4), (41:5)
in 41.3.2.; (41:8), (41:9), (41:10) in 41.4.2.; and our (44:4), (44:5), (44:6), (44:7).

DECOMPOSABLE GAMES

361

44.3. Composition and Decomposition of Solutions.

The Main Possibilities and Surmises

44.3.1. Let V/i W* be two solutions for the games A, H respectively.
Their composition yields an imputation set U/ which one might expect
to be a solution for the game T. Indeed, U/ is the expression of a standard
of behavior which can be formulated as follows. We give the verbal formula-
tion in the text under (44:B:a)-(44:B:c), stating the mathematical equiva-
lents in footnotes, which, as the reader will verify, add up precisely to our
definition of composition.

(44:B:a) The players of J always obtain together exactly their

“just dues” (zero), and the same is true for the players of K. 1

There is no connection whatever between the fate of play-
ers in the set J and in the set K . 2 *

The fate of the players in J is governed by the standard of
behavior VO, 3 the fate of the players in K is governed by the
standard of behavior W*. 4

If the two constituent games are imagined to occur absolutely separate
from each other, then this is the plausible way of viewing their separate
solutions V/, W* as one solution U/ of the composite game T.

However, since a solution is an exact concept, this assertion needs a
proof. I.e. we must demonstrate this:

(44:C) If V/, W k are solutions of A, H, then their composition U/ is

a solution of V.

44.3.2. This, by the way, is another instance of the characteristic
relationship between common sense and mathematical rigour. Although
an assertion (in the present case that U/ is a solution whenever Vj, W* are)
is required by common sense, it has no validity within the theory (in this
case based on the definitions of 30.1.1.) unless proved mathematically. To
this extent it might seem that rigour is more important than common sense.
This, however, is limited by the further consideration that if the mathe-
matical proof fails to establish the common sense result, then there is a
strong case for rejecting the theory altogether. Thus the primate of the
mathematical procedure extends only to establish checks on the theories —
in a way which would not be open to common sense alone.

1 Every element a / of U/ is decomposable.

— > — >

* Any 0 j which is used in forming U / and any y which is used in forming U/, give

by composition an element a i of U/.

* The above mentioned 0 / are precisely the elements of V/.

4 The above mentioned y k are precisely the elements of W *.

(44:B:b)

(44:B:c)

362 COMPOSITION AND DECOMPOSITION OF GAMES

It will be seen that (44 :C) is true, although not trivial.

One might be tempted to expect that the converse of (44 :C) is also
true, i.e. to demand a proof of this:

(44 :D) If U/ is a solution of T, then it can be decomposed into solu-

tions V/, Wk of A, H.

This is prima facie quite plausible: Since T is the composition of what
are for all intents and purposes two entirely separate -games, how could any
solution of T fail to exhibit tjiis composite structure?

The surprising fact is, however, that (44 :D) is not true in general.
The reader might think that this should induce us to abandon — or at least
to modify materially — our theory (i.e. 30.1.1.) if we take the above method-
ological statement seriously. Yet we will show, that the “common sense”
basis for (44 :D) is quite questionable. Indeed, our result, contradicting
(44 :D) will provide a very plausible interpretation — which connects it
successfully with well known phenomena in social organizations.

44 . 3 . 3 . The proper understanding of the failure of (44 :D) and of the
validity of the theory which replaces it, necessitates rather detailed con-
siderations. Before we enter upon these, it might be useful to make, in
anticipation, some indications as to how the failure of (44 :D) occurs.

It is natural, to split (44 :D) into two assertions:

(44:D:a) If U/ is a solution of T, then it is decomposable (for J, K).

(44:D:b) If a solution U/ of T is decomposable (for /, K), then its

constituents V/, W k are solutions for A, H.

Now it will appear that (44:D:b) is true, and (44:D:a) is false. I.e.
it can happen that a decomposable game r possesses an indecomposable
solution. 1

However, the decomposability of a solution (or of any set of imputations)
is expressed by (44:B:a)-(44:B:c) in 44.3.1. So one or more of these condi-
tions must fail for the indecomposable solution referred to above. Now
it will be seen (cf. 46.11.) that the condition which is not satisfied is (44:B:a).
This may seem to be very grave, because (44:B:a) is the primary condition
in the sense that when it fails, the conditions (44:B:b), (44:B:c) cannot
even be formulated.

The concept of decomposition possesses a certain elasticity. This
appeared_in 42.2.1., 42.2.2. and 42.5.2., where we succeeded in ridding
ourselves of an inconvenient auxiliary condition connected with the decom-
posability of a game by modifying that concept. It will be seen that our
difficulties will again be met by this procedure — so that (44 :D) will be
replaced by a correct and satisfactory theorem. Hence we must aim at
modifying our arrangements, so that the condition (44:B:a) can be discarded.

We will succeed in doing this, and then it will appear that conditions
(44:B:b), (44:B:c) make no difficulties and that a complete result can be
obtained.

1 This is similar to the phenomenon that a symmetric game may possess an asym-
metric solution. Cf. 37.2.1.

DECOMPOSABLE GAMES

363

44.4. Extension of the Theory. Outside Sources

44 . 4 . 1 . It is now time to discard the normalization which we introduced
(temporarily) in 44.1.: That the games under consideration are zero-sum.
We return to the standpoint of 42.2.2. according to which the games are
constant-sum.

These being understood, consider a game V which is decomposable
(for J , K) with J-, ^-constituents A, H.

The theory of composability and decomposability of imputations, as
given in 44.2.1., 44.2.2. could now be repeated with insignificant changes.
(44:l)-(44:4) may be taken over literally, while (44:5)-(44:7) are only
modified in their right hand sides. Since (30:2) of 30.1.1. has been replaced
by (42:8*) of 42.4.1. those formulae (44:5)-(44:7) now become:

(44:5*)

(44:6*)

and

(44:7*)

£ = v(j),

»'«r

£ «r-v(X),

1 "

X +

i"

X

= v(Z) - v(J) + v(K).

(The last equation on the right hand side by (42:6:b) in 42.3.2., or equally
by (41:6) in 41.3.2. with S = /, T = K.) The situation is exactly as in
44.2.1., indeed, it really arises from that one by the isomorphism of 42.4.2.

Thus a i fulfills (44:7*), but for its decomposability (44:5*), (44:6*) are
needed — and (44:7*) does imply the equivalence of (44:5*) and (44:6*),
but it fails to imply the validity of either.

So the criterion of decomposability (44: A) in 44.2.1. is again true, only
with our (44:5*), (44:6*) in place of its (44 :5), (44:6). And the final con-
clusion of 44.2.2. may be repeated: Decomposition of an a j (into p /, y K )
is possible if and only if the two self contained sets of players J, K are

given by this imputation a / precisely their just dues — which are now v(J),

v(K).'

Since we know that this limitation of the decomposability of imputa-
tions — the reason for (44:B:a) in 44.3.1. — is a source of difficulties, we have
to remove it. This means removal of the conditions (44:5*), (44:6*), i.e.
of the condition (42:8*) in 42.4.1. from which they originate.

44 . 4 . 2 . According to the above, we will attempt to work the theory of a
constant-sum game r with a new concept of imputations, which is based on
(42:7) of 42.4.1. (i.e. on (30:1) of 30.1.1.) alone, without (42:8*) in 42.4.1.
In other words 2

1 Instead of zero, as loc. cit.

* We again denote the players by 1, • • • , n.

364 COMPOSITION AND DECOMPOSITION OF GAMES

An extended imputation is a system of numbers ai, ■ ■ • , a n with this
property:

(44:8) on k v((i)) for i = 1, • • • , n.

n

We impose no conditions upon ^ a»*. We view these extended imputa-

»-i

tions, too, as vectors

= {<* 1 > * * * >

44 . 4 . 3 . It will now be necessary to reconsider all our definitions which are
rooted in the concepts of imputation — i.e. those of 30.1.1. and 44.2.1. But,
before we do this, it is well to interpret this notion of extended imputations.

The essence of this concept is that it represents a distribution of certain
amounts between the players, without demanding that they should total
up to the constant sum of the game r.

Such an arrangement would be extraneous to the picture that the players
are only dealing with each other. However, we have always conceived of
imputations as a distributive scheme proposed to the totality of all players.
(This idea pervades, e.g. all of 4.4., 4.5.; it is quite explicit in 4.4.1.) Such
a proposal may come from one of the players, 1 but this is immaterial. We
can equally imagine, that outside sources submit varying imputations to
the consideration of the players of T. All this harmonizes with our past
considerations, but in all this, those “ outside sources” manifested them-
selves only by making suggestions — without contributing to, or withdrawing
from, the proceeds of the game.

44.5. The Excess

44 . 5 . 1 . Now our present concept of extended imputations may be taken
to express that the “ outside sources” can make suggestions which actually
involve contributions or withdrawals, i.e. transfers. For the extended

imputation a = {an, • • * , a„} the amount of this transfer is

n

(44:9) e = ^ on - v(/)

i-1
— ►

and will be called the excess of a . Thus

e > 0 for a contribution,

(44:10) e = 0 if no transfer takes place,

e < 0 for a withdrawal.

1 Who tries to form a coalition. Since we consider the entire imputation as his
proposal, this necessitates our assuming that he is even making propositions to those
players, who will not be included in the coalition. To these he may offer their respective
minima v((i)) (possibly more, cf. 38.3.2. and 38.3.3.). There may also be players in
intermediate positions “between included and excluded” (cf. the second alternative in
37.1.3.). Of course, those less favored players may make their dissatisfaction effective,
this leads to the concept of domination, etc.

DECOMPOSABLE GAMES

365

It will be necessary to subject this to certain suitable limitations, in
order to obtain realistic problems; and we will take due account of this.

It is important to realize how these transfers interact with the game.
The transfers are part of the suggestions made from outside, which are
accepted or rejected by the players, weighed against each other, according
to the principles of domination, etc. 1 In the course of this process, any
dissatisfied set of players may fall back upon the game T, which is the
sole criterion of the effectivity of their preference of their situation in one
(extended) imputation against another. 2 Thus the game, the physical
background of the social process under consideration, determines the
stability of all details of the organization — but the initiative comes through
the outside suggestions, circumscribed by the limitations of the excess
referred to above.

44 . 5 . 2 . The simplest form that this “limitation” of the excess can take,
consists in prescribing its value e explicitly. In interpreting this prescrip-
tion, (44:10) should be remembered.

The situation which exists when e ^ 0 may at first seem paradoxical.

This is particularly true when e < 0, i.e. when a withdrawal from outside
is attempted. Why should the players, who could fall back on a game of
constant sum v(/) accept an inferior total? I.e. how can a “standard of
behavior,” a “social order,” based on such a principle be stable? There is,
nevertheless an answer: The game is only worth v(7) if all players form a
coalition, act in concert. If they are split into hostile groups, then each
group may have to estimate its chances more pessimistically and such a
division may stabilize totals that are inferior to v(J). 3

The alternative e > 0, i.e. when the outside interference consists of a
free gift, may seem less difficult to accept. But in this case too, it will be

1 This is, of course, a narrow and possibly even somewhat arbitrary description of the
social process. It should be remembered, however, that we use it only for a definite and
limited purpose: To determine stable equilibria, i.e. solutions. The concluding remarks
of 4.6.3. should make this amply clear.

2 We are, of course, alluding to the definitions of effectivity and domination, cf. 4.4.1.
and the beginning of 4.4.3. — given in exact form in 30.1.1. We will extend the exact
definitions to our present concepts in 44.7.1.

3 For a first quantitative orientation, in the heuristic manner: If the players are
grouped into disjunct sets (coalitions) Si, • • • , <S P , then the total of their own valua-
tions is v(Si) + • • • + v(S p ). This is ^ v(7) by (42:6:c) in 42.3.2.

Oddly enough, this sum is actually =* v(7) when p — 2 by (42:6:b) in 42.3.2. —
i.e. in this model the disagreements between three or more groups are the effective sources
of damage.

Clearly by (42:6:c) in 42.3.2. the above sums v(S 0 + • • • + v(S p ) are all

n

^ ^ v ( (t ) ) . On the other hand, this latter expression is one of them (put p — n,
<- 1

S x — (i)). So the damage is greatest when each player is isolated from all others.

n

The whole phenomenon disappears, therefore, when ^ v((i)) =* v(7) , i.e. when the
game is inessential. (Cf. (42:11) in 42.5.1.)

366 COMPOSITION AND DECOMPOSITION OF GAMES

necessary to study the game in order to see how the distribution of this
gift among the players can be governed by stable arrangements. It has to
be expected, that the optimistic appraisal of their own chances, derived
from the possibilities of the various coalitions in which they might par-
ticipate will determine the players in making their claims. The theory
must then provide their adjustment to the available total.

44.6. Limitations of the Excess.

The Non-isolated Character of a Game in the New Setup

44 . 6 . 1 . These considerations indicate that the excess e must be neither
too small (when e < 0), nor too large (when e > 0). In the former case a
situation would arise where each player would prefer to fall back on the
game, even if the worst should happen, i.e. if he has to play it isolated. 1
In the latter case it will happen that the “free gift” is “too large,” i.e. that
no player in any imagined coalition can make such claims as to exhaust the
available total. Then the very magnitude of the gift will act as a dissolvent
on the existing mechanisms of organizations.

We will see in 45. that these qualitative considerations are correct
and we will get from rigorous deductions the details of their operation and
the precise value of the excess at which they become effective.

44 . 6 . 2 . In all these considerations the game T can no longer be con-
sidered as an isolated occurrence, since the excess is a contribution or a
withdrawal by an outside source. This makes it intelligible that this
whole train of ideas should come up in connection with the decomposition
theory of the game T. The constituent games A, H are indeed no longer
entirely isolated, but coexistent with each other. 2 Thus, there is a good
reason to look at A, H in this way — whether the composite game T should
be treated in the old manner (i.e. as isolated), or in the new one, may be
debatable. We shall see, however, that this ambiguity for T does not
influence the result essentially., whereas the broader attitude concerning A,
H proves to be absolutely necessary (cf. 46.8.3. and also 46.10.).

When a game T is considered in the above sense, as a non-isolated
occurrence, with contributions or withdrawals by an outside source, one
might be tempted to do this: Treat this outside source also as a player,
including him together with the other players into a larger game T'. The
rules of T' (which includes T) must then be devised in such a manner as to
provide a mechanism for the desired transfers. We shall be able to meet
this demand with the help of our final results, but the problem has some
intricacies that are better considered only at that stage.

n

1 This happens when the proposed total v(/) + e is < ^ v((i)). As the last expres-

* — l

sion is equal to v(/) — ny (by (42:11) in 42.5.1.) this means e < —ny.

We will see in 45.1. that this is precisely the criterion for e being “too small/'

* This in spite of the absence of “interactions," as far as the rules of the game are
concerned; cf. 41.2.3., 41.2.4.

DECOMPOSABLE GAMES

367

44.7. Discussion of the New Setup E(e 0 ), F(e 0 )

44.7.1. The reconsideration of our old definitions mentioned at the
beginning of 44.4.3. is a very simple matter.

For the extended imputations we have the new definitions of 44.4.2.
The definitions of effectivity and domination we take over unchanged from
30.1.1. 1 — the supporting arguments brought forward in the discussion which
led up to those definitions appear to lose no strength by our present general-
izations. The same applies to our definition of solutions eod. 2 with one
caution: The definition of a solution referred to makes the concept of a
solution dependent upon the set of all imputations in which it is formed.
Now in our present setup of extended imputations we shall have to consider
limitations concerning them — notably concerning their excesses — as indi-
cated in 44.5.1. These restrictions will determine the set of all extended
imputations to be considered and thereby the concept of a solution.

44.7.2. Specifically we shall consider two types of limitations.

First, we shall consider the case where the value of the excess is pre-
scribed. Then we have an equation

(44:11) e = e 0

with a given e 0 . The meaning of this restriction is that the transfer from
outside is prescribed, in the sense of the discussion of 44.5.2.

Second, we shall consider the case where only an upper limit of the excess
is prescribed. Then we have an inequality

(44:12) e ^ e Q

with a given e Q . The meaning of this restriction is that the transfer from
outside is assigned a maximum (from the point of view of the players who
receive it).

The case in which we are really interested is the first one, i.e. that one
of 44.5.2. The second case will prove technically useful for the clarification
of the first one — although its introduction may at first seem artificial.
We refrain from considering further alternatives because we will be able to
complete the indicated discussion with these two cases alone.

Denote the set of all extended imputations fulfilling (44:11) (first case)
by E(e 0 ). Considering (44:9) in 44.5.1., we can write (44:11) as

n

(44:11*) 2a< = v(/)+e„.

>- 1

Denote the set of all extended imputations fulfilling (44:12) (second case)
by F(e 0 ). Considering (44:9) in 44.5.1., we can write (44:12) as

(44:12*) f aig v(/) + e 0 .

»- i

1 I.e. (30:3); (30:4:a)-(30:4:c) loc. cit., respectively.

* I.e. (30:5 :a), (30:5:b) or (30:5:c) eod.

368 COMPOSITION AND DECOMPOSITION OF GAMES

For the sake of completeness, we repeat the characterization of an extended
imputation which must be added to (44:11*), as well as to (44:12*):

(44:13) on ^ v((t)), for i = 1, • • • , n.

Note that the definitions of (44:9) as well as (44:11*), (44:12*) and (44:13)
are invariant under the isomorphism of 42.4.2.

44.7.3. Now the definition of a solution can be taken over from 30.1.1.
Because of the central role of this concept we restate that definition, adjusted
to the present conditions. Throughout the definition which follows, E{e 0 )
can be replaced by F(e 0 ) y as indicated by [ ].

A set V £ E(e o) [F(e 0 )l is a solution for E(co) [F(e 0 )] if it possesses the follow-
ing properties:

(44:E:a) No 0 in V is dominated by an a in V.

(44:E:b) Every 0 of E(eo)[F(e 0 )] not in V is dominated by some a

in V.

(44:E:a) and (44:E:b) can be stated as a single condition:

(44:E:c) The elements of V are those elements of E(e 0 ) [F(e 0 )] which

are undominated by any element of V.

It will be noted that E( 0) takes us back to the original 30.1.1. (zero-sum
game) and 42.4.1. (constant-sum game).

44.7.4. The concepts of composition , decomposition and constituents
of extended imputations can again be defined by (44:l)-(44:4) of 44.2.1.
As pointed out in 44.4.2. the technical purpose of our extending the concept
of imputation is now fulfilled. Decomposition as well as composition can
now always be carried out.

The connection of these concepts with the sets E(e 0 ) and F(e 0 ) is not so
simple; we will deal with it as the necessity arises.

For the composition , decomposition and constituents of sets of extended
imputations the definitions of 44.2.3. can now be repeated literally.

45. Limitations of the Excess. Structure of the Extended Theory
45.1. The Lower Limit of the Excess

45.1. In the setups of 30.1.1. and of 42.4.1. imputations always existed.
It is now different: Either set E(e 0 ), F(e 0 ) may be empty for certain e 0 .
Obviously this happens when (44:11*) or (44:12*) of 44.7.2. conflict with
(44:13) eodem — and this is clearly the case for

v(7) + e 0 < £ v((0)

»= 1

in both alternatives. As the right hand side is equal to v(l) — ny by
(42:11) in 42.5.1., this means

(45:1)

e 0 < — ny

STRUCTURE OF THE EXTENDED THEORY

369

If E(e<>) [F(e o)] is empty, then the empty set is clearly a solution for it —
and since it is its only subset, it is also its only solution. 1 If, on the other
hand, E(e 0 ) [F(e 0 )] is not empty, then none of its solutions can be empty.
This follows by literal repetition of the proof of (31 :J) in 31.2.1.

The right hand side of the inequality (45:1) is determined by the game
T; we introduce this notation for it (with the opposite sign, and using (42:11)
in 42.5.1.):

(45:2) |r|, = ny = v(J) - £ v((i)).

*- 1

Now we can sum up our observations as follows:

(45: A) If

eo < - |r|i,

then E(e 0 ), F(e 0 ) are empty and the empty set is their only
solution. Otherwise neither E(e 0 ) nor F(e 0 ) nor any solution of
either can be empty.

This result gives the first indication, that “too small” values of e 0
(i.e. e) in the sense of 44.6.1. exist. Actually, it corroborates the quanti-
tative estimate of footnote 1 on p. 366.

46.2. The Upper Limit of the Excess. Detached and FuUy Detached Imputations

45 . 2 . 1 . Let us now turn to those values of e 0 (i.e. e ), which are “too
large” in the sense of 44.6.1. When does the disorganizing influence of
the magnitude of e, which we there foresaw, manifest itself?

As indicated in 44.6.1., the critical phenomenon is this: The excess
may be too large to be exhausted by the claims which any player in any
imagined coalition can possibly make. We proceed to formulate this idea
in a quantitative way.

— >

It is best to consider the extended imputations a themselves, instead

of their excesses e. Such an a is past any claims which may be made in
any coalition, if it assigns to the players of each (non-empty) set S £ I
more than those players could get by forming a coalition in T, i.e. if

(45:3) X a» > v(S) for every non-empty set S £ I.

i in 8

Comparing this with (30:3) in 30.1.1. shows that our criterion amounts to

demanding that every non-empty set S be ineffective for a .

In our actual deductions it will prove advantageous to widen (45:3)
somewhat by including the limiting case of equality. The condition then
becomes

1 In spit© of its triviality, this circumstance should not be overlooked. The text
actually repeats footnote 2 on p. 278.

370 COMPOSITION AND DECOMPOSITION OF GAMES

(45:4) X gi v(S) for every Ssl. 1

i in S

It is convenient to give these a a name. We call the a of (45:3) fully
detached , and those of (45:4) detached. As indicated, the latter concept
will be really needed in our proofs — both termini are meant to express that
the extended imputation is detached from the game, i.e. that it cannot be
effectively supported within the game by any coalition.

45 . 2 . 2 . One more remark is useful:

The only restriction imposed upon extended imputations is (44:13) of

44.7.2.:

(45:5) a t ^ v((i)) for i = 1, • • • , n.

Now if the requirement (45:4) of detachedness is fulfilled — and hence
a fortiori if the requirement (45:3) of full detachedness is fulfilled — then it is
unnecessary to postulate the condition (45:5) as well. Indeed, (45:5)
is the special case of (45:4) for S = (i).

This remark will be made use of implicitly in the proofs which follow.

45 . 2 . 3 . Now we can revert to the excesses, i.e. characterize those which
belong to detached (or fully detached) imputations. This is the formal
characterization:

(45 :B) The game T determines a number |r | 2 with the following

properties:

(45:B:a) A fully detached extended imputation with the excess e

exists if and only if

e > |r| 2 .

(45:B:b) A detached extended imputation with the excess e exists

if and only if

e ^ |r | 2 . 2

>

Proof: Existence of a detached a 3 : Let a 0 be the maximum of all v(S) f

S Si (so a 0 § v(Q) = 0). Put a 0 = {aj, * • • , a°) = {a 0 , • • • , a 0 }.
Then for every non-empty S s I we have ^ a? g: a 0 ^ v(£). This is

» in 8

(45:4), so a 0 is detached.

1 It is no longer necessary to exclude 5=0, since (45:4) unlike (45:3) is true when
5—©. Indeed, then both sides vanish.

* The intuitive meaning of these statements is quite simple: It is plausible that in
order to produce a detached or a fully detached imputation, a certain (positive) minimum
excess is required. |r| 2 is this minimum, or rather lower limit. Since the notions
“detached” and “fully detached ,, differ only in a limiting case (the « sign in (45:4)),
it stands to reason that their lower limits be the same. These things find an exact
expression in (45 :B).

* Note that it is necessary to prove this ! The evaluation which we give here is crude,
for more precise ones cf. (45 :F) below.

STRUCTURE OF THE EXTENDED THEORY

371

Properties of the detached a : According to the above, detached

a - {c*i, •••,«„)

n

exist, and with them their excesses e = ^ <*< — v(J). By (45:4) (with

»-i

S = 7) all these e are 0. Hence it follows by continuity, that these e have

a minimum e *. Choose a detached a * = {af, • • • , a*) with this excess
e*. 1

We now put

(45:6) |r|t = «*.

— i

Proof of (45:B:a), (45:B :b):If a = {ai, • • • , a n } is detached, then by

n .

definition e = ^ a* — v(7) ^ e*. If a = {ai, ••*,«,»} is fully detached,
»- 1

then (45:3) remains true if we subtract a sufficiently small 6 > 0 from
each So a' = {a t — 5, • • • , a n — 5} is detached. Hence by defini-

n

tion e — n5 = £ (a< — 5) — v(/) ^ e*, e > e*.

»-i

— >

Consider now the detached a * = {a*, • • , a*} with

l a* - v(I) = e*

1-1

Then (45:4) holds for a*; hence (45:3) holds if we increase each a* by a
5 > 0. So a " = { af -f 6, • • • , a* + 5 ) is fully detached. Its excess is

n

e = Yf ( a i + 5) — v(7) = e* + n5. So every e = e* + n$, 8 > 0, i.e.
i- 1

every e > e *, is the excess of a fully detached imputation — hence a fortiori

of a detached one; and e* is, of course, the excess of a detached imputation
— ►
a *.

Thus all parts of (45:B:a), (45:B:b) hold for (45:6).

45 . 2 . 4 . The fully detached and the detached extended imputations are
also closely connected with the concept of domination. The properties
involved are given in (45 :C) and (45 :D) below. They form a peculiar anti-
thesis to each other. This is remarkable, since our two concepts are strongly
analogous to each other — indeed, the second one arises from the first one by
the inclusion of its limiting cases.

1 This continuity argument is valid because the =* sign is included in (45:4).

372 COMPOSITION AND DECOMPOSITION OF GAMES

(45 :C) A fully detached extended imputation a dominates no other

— ►

extended imputation p .

— > — > — >

Proof: If a H P f then a must possess a non-empty effective set.

— ►

(45 :D) An extended imputation a is detached if and only if it is

dominated by no other extended imputation p .

Proof: Sufficiency of being detached :Let a = {<*i, • • • , a n ) be detached.

Assume a contrario p fr* a , with the effective set S. Then S is not empty;
a» < Pi for i in S. So £ a, < ^ 0* ^ v(S) contradicting (45:4).

t in 8 i in S

Necessity of being detached: Assume that a = {a lf • • • , a n ] is not
detached. Let S be a (necessarily non-empty) set for which (45:4) fails,
i.e. ^ a* < v{S). Then for a sufficiently small 5 > 0, even

i in 3

2 («i + «) s v(s).

t in S

Put i9 = {0i, • • • , 0n} = {ai + 6, ■ • • , a, + 5j, then always o< < 0«
and S is effective for 0 : £ 0< g v(<S). Thus 0 h a .

tin s

46.3. Discussion of the Two Limits |r| lt 1'iL Their Ratio

46 . 3 . 1 . The two numbers |r|i and |r| 2 , as defined in (45:2) of 45.1.
and in (45 :B) of 45.2.3. are both in a way quantitative measures of the
essentiality of T. More precisely :

(45 :E) If T is inessential, then |r|i = 0, [r|* = 0.

If T is essential, then |r|! > 0, |r| 2 > 0.

Proof: The statements concerning |r|i, which is = ny by (45:2) of 45.1.,
coincide with the definitions of inessentiality and essentiality of 27.3., as
reasserted in 42.5.1.

The statements concerning |r| a follow from those concerning |r|i, by
means of the inequalities of (45 :F), which we can use here.

45 . 3 . 2 . The quantitative relationship of |T|i and |r| 2 is charactenzed as
follows:

Always

— |r|. £ |r|, £ n -^ |r|,.

(45 :F)

n — 1

2

STRUCTURE OF THE EXTENDED THEORY

373

Proof: As we know, |T| i and |r|* are invariant under strategic equivalence,
hence we may assume the game r to be zero-sum, and even reduced in the
sense of 27.1.4. We can now use the notations and relations of 27.2.

Since |r|i = ny, we want to prove that

(45:7) ^yg|r|, g n(n ~ 2) r

Proof of the first inequality of (45:7): Let a = {a 1( •••,«») be
detached. Then (45:4) gives for the (n — l)-element set S = I — (k),

n

£ a < - v (£) = y> 5 - e -

t-1 tin 8

(45:8) ^ a< — a* ^ 7 .

n n

Summing (45:8) over k = 1, • • • , n, gives n a* ny,

i-i

*-i

n n n

i.e. (n - 1 ) £ a< ^ J 17 , £ on £ ” . y. Now v(7) = 0, so e = £

i»l |«1 /tl

Thus e ^ r 7 for all detached imputations; hence |F|» ^ T 7 .

Proof of the second inequality of (45 :7):

Put a 0# = 1 ^ 7 , and = (a?», • • • , a™) = {a M , • • • , a 09 }.

This a 00 is detached, i.e. it fulfills (45:4) for all Ss/. Indeed: Let p be
the number of elements of S. Now we have:
p = 0: S = ©, (45:4) is trivial.
p a* 1 : S = (i), (45:4) becomes a 00 ^ v((i)),

i.e. — 7 ^ — 7 which is obvious.

&

p ^ 2: (45:4) becomes pa 00 ^ v(<S), but by (27:7) in 27.2.

v(S) g (n - p)y,

^ 2

so it suffices to prove pa 00 ^ (n — p )7 i.e. p — ^ — 7 ^ (n — p) 7 . This
n

amounts to p 75 7 ^ n 7 , which follows from p ^ 2 .

— ►

Thus a 00 is indeed detached. As v(/) = 0, the excess is

*!L=J) 7 .

Hence |r|» S n ^ n g t-

374 COMPOSITION AND DECOMPOSITION OF GAMES

45 . 3 . 3 . It is worth while to consider the inequalities of (45 :F) for
n = 1,2, 3, 4, • • • successively:

n = 1,2: In these cases the coefficient — — T of the lower bound of the
’ n — 1

Tt - 2

inequality is greater than the coefficient — ^ u PP er bound., 1 This

may seem absurd. But since T is necessarily inessential for n = 1,2 (cf.
the first remark in 27.5.2.), we have in these cases |r|i = 0, |r|» = 0, and
so the contradictions disappear.

1 tx 2

n = 3: In this case the two coefficients T and — s — coincide:

n — 1 2

Both are equal to So the inequalities merge to an equation:

(45:9) |r|* = i|r|i.

n ^ 4: In these cases the coefficient — — - of the lower bound is definitely

n — 1

Tt — 2

smaller than the coefficient — ^ — of the upper bound. 2 * * * * * So now the inequal-
ities leave a non- vanishing interval open for |r 2 |.

The lower bound |r| 2 = ~ZT\ l^li 18 P recise > i e - there exists for each
n ^ 4 an essential game for which it is assumed. There also exist for each
n ^ 4 essential games with |r| 2 > — - |r|i, but it is probably not pos-

Tt 2

sible to reach the upper bound of our inequality, |r| 2 = — ^ — |r|i- The

precise value of the upper bound has not yet been determined. We do not
need to discuss these things here any further. 8

45 . 3 . 4 . In a more qualitative way, we may therefore say that |r|i,|r| 2
are both quantitative measures of the essentiality of the game I\ They
measure it in two different, and to a certain extent, independent ways.
Indeed, the ratio |r| 2 /|r|i, which never occurs for n = 1,2 (no essential
games!), and is a constant for n = 3 (its value is £), is variable with T for
each n ^ 4.

We saw in 45.1., 45.2., that these two quantities actually measure the
limits, within which a dictated excess will not “disorganize” the players,
in the sense of 44.6.1. Judging from our results, an excess e < — | T| i is
“too small” anti an excess e > |r| 2 is “too great” in that sense. This
view will be corroborated in a much more precise sense in 46.8.

1 They are «, — \ for n — 1; 1, 0 for n * 2. Note also the paradoxical values « and

-*!

1 n - 2

*- r < — ^ — means 2 < (n — l)(n — 2) which is clearly the case for all n £ 4.

71 — 1 L

9 For n — 4 our inequality is } |r|i 5 |r| 2 £ |r|i. As mentioned above, we know an

essential game with |r| 2 - J |f|i and also one with |F|* - | |r|}.

STRUCTURE OF THE EXTENDED THEORY

375

45.4. Detached Imputations and Various Solutions.

The Theorem Connecting E(e 0 ), F(e 0 )

45 . 4 . 1 . (44:E:c) in the definition of a solution in 44.7.3. and our result
(45 :D) in 45.2.4. give immediately:

(45 :G) A solution V for E(e 0 ) [F(e 0 )] must contain every detached

extended imputation of E(e 0 ) [F(e 0 )].

The importance of this result is due to its role in the following considera-
tion.

After what was said at the beginning of 44.7.2. about the roles of 2?(e 0 )
and F(e o), the importance of establishing the complete inter-relationship
between these two cases will be obvious. I.e. we must determine the
connection between the solutions for E(e 0 ) and F(e 0 ).

Now the whole difference between E(e 0 ) and F(e 0 ) and their solutions
is not easy to appraise in an intuitive way. It is difficult to see a priori
why there should be any difference at all: In the first case the “gift,” made
to the players from the outside, has the prescribed value e 0 , in the second
case it has the prescribed maximum value e 0 . It is difficult to see how the
“outside source,” which is willing to contribute up to e 0 can ever be allowed
to contribute less than e 0 in a “stable” standard of behavior (i.e. solution).
However, our past experience will caution us against rash conclusions in this
respect. Thus we saw in 33.1. and 38.3. that already three and four-person
games possess solutions in which an isolated and defeated player is not
“exploited” up to the limit of the physical possibilities — and the present
case bears some analogy to that.

45 . 4 . 2 . (45 :G) permits us to make a more specific statement:

A detached extended imputation a belongs by (45 :G) to every solution for

F(e 0 ), if it belongs to F(e 0 ). On the other hand, a clearly cannot belong to
any solution for 2?(e 0 ) if it does not belong to E(eo). We now define:

(45:10) D*(e 0 ) is the set of all detached extended imputations a in

F(e 0 ), but not in E(e 0 ).

So we see: Any solution of F(e o) contains all elements of D*(e o); any solu-
tion of E(e 0 ) contains no element of D*(e 0 ). Consequently F(e 0 ) and E(e 0 )
have certainly no solution in common if D*(e 0 ) is not empty.

— ►

Now the detached a of D*(e 0 ) are characterized by having an excess
e ^ eo, but not c = eo — i.e. by

(45:11) e < e 0 .

From this we conclude:

(45 :H) D*(e 0 ) is empty if and only if

eo ^ |r| s .

376 COMPOSITION AND DECOMPOSITION OF GAMES

Proof : Owing to (45 :B) and to (45:11) above, the non-emptiness of
D*(eo) is equivalent to the existence of an e with |r|* g e < e 0 — i.e. to
e 0 > |r| t . Hence the emptiness of D*(e 0 ) amounts to e 0 g |r|*.

Thus the solutions for F(e 0 ) andfor E{e^) are sure to differ, whene 0 > |r|*.
This is further evidence that e 0 is “too large” for normal behavior when it is

> |r|t.

45 . 4 . 3 . Now we can prove that the difference indicated above is the only
one between the solutions for E(e 0 ) and for F(e 0 ). More precisely:

(45:1) The relationship

(45:12) V^W = V u D*(e 0 )

establishes a one-to-one relationship between all solutions V
for E(e o) and all solutions W for F(e 0 ).

This will be demonstrated in the next section.

45.5. Proof of the Theorem

45 . 5 . 1 . We begin by proving some auxiliary lemmas.

The first one consists of a perfectly obvious observation, but of wide
applicability :

(45 :J) Let the two extended imputations y = { 71 , • • • , y n ) and

8 =■ {$ 1 , • • * , 5 n } bear the relationship
(45:13) y, ^ 5, for all i = 1, • • • , n\

then for every a , a H 7 implies a H 8 .

The meaning of this result is, of course, that (45:13) expresses some

kind of inferiority of 5 to 7 — in spite of the intransitivity of domination.
This inferiority is, however, not as complete as one might expect. Thus

one cannot make the plausible inference of 7 h p from 5 H 0 , because the

effectivity of a set S for 8 may not imply the same for 7 . (The reader
should recall the basic definitions of 30.1.1.)

It should also be observed, that (45 :J) emerges only because we have
extended the concept of imputations. For our older definitions (cf. 42.4.1.)

♦ n n

we would have had £ 7* = £ $*; hence y { ^ for all i = 1, • • • , n

*- 1 t-i

necessitates 7 < = 5* for all i = 1 , • • • , n, i.e. 7 = $ .

45 . 5 . 2 . Now four lemmas leading directly to the desired proof of (45:1).

If a H p with a detached and in F(e 0 ) and 0 in JE(co), then
there exists an a ' H 0 with a ' detached and in E(e 0 ).

(45 :K)

STRUCTURE OF THE EXTENDED THEORY 377

Proof: Let 5 be the set of (30:4:a)-(30:4:c) in 30.1.1. for the domination
— > — ►

ct H P . S = / would imply > ft for all i = 1, • • • , n so
£ «< - v(7) > S ft - v(/).

But as a is in F(e«) and 0 in E(e 0 ), so £ — v(/> ^ e 0 = £ ft — v(7),

t - 1 » - 1

contradicting the above.

So S ^ I. Choose, therefore, an t 0 = 1, ■ • • , n, not in S. Define
<*' = Wu ••’,<} with

< = «<, + *,
a[ = a, for i i 0 ,

" — >

choosing so that £ a' — v(7) = e 0 . Thus all <*(• ^ a,-; hence a '

»-l

is detached and it is clearly in E(e 0 ). Again, as a[ == a* for i 5* t‘o, hence
for all i in S , so our a p implies a ' H 0 .

(45:L) Every solution W for F(e 0 ) has the form (45:12) of (45:1)

for a unique V £ E(e o). 1

Proof : Obviously the V in question — if it exists at all — is the intersection
W n E(e 0), so it is unique. In order that (45:12) should hold for

V = Wn E(e 0 ),

we need only that the remainder of W be equal to D*(e 0 ), i.e.

(45:14) W - E(e 0 ) = D*(e 0).

Let us therefore prove (45:14).

Every element of D*(e 0) is detached and in F(e 0) — so it is in W by (45 :G).
Again, it is not in E(e 0 ), so it is in W — E(e 0 ). Thus

(45:15) W - E(e 0 ) 2 D*(e 0 ).

If also

(45:16) W - B(e 0) £ D*(e 0 ),

then (45:15), (45:16) together give (45:14), as desired. Assume therefore,
that (45:16) is not true.

Accordingly, consider an a = {ai, • • • , a n | in W - E(e 0) and not

n

in D*(e 0 ). Then a is in F(e 0 ), but not in E(c 0 ), so £ a< — v(7) < e 0 . As

1 We do not yet assert that this V is a solution for E(e %) — that will come in (45:M).

378 COMPOSITION AND DECOMPOSITION OF GAMES

a is not in D*(e 0 ) ) this excludes its being detached. Hence there exists a
non-empty set S with ^ a* < v(S).

Now form a ' = {aj, • • • , o

ot[ = Cti +

for i in S y
for i not in S,

choosing € > 0 so that still ^ a[ — v(/) ^ e 0 and a'- ^ v(<S). So a '

t « 1 * in S

is in F(co). If it is not in W> then (as W is a solution for F(e 0 )) there exists

a P in W with p H a'. As all a' ^ a*, this implies (3 H a by (45 :J).

— > — > . — ►

This is impossible, since both /3 , a belong to (the solution) W. Hence a '

must be in W. Now a< > a, for all i in S, and £ — v (^)- So

» in <S

a ' H a . But as both a ', a belong to (the solution) W, this is a contra-
diction.

(45 :M) The V of (45 :L) is a solution for E(e 0 ).

Proof : V C E(e 0 ) is clear, and V fulfills (44:E:a) of 44.7.3. along with W
(which is a solution for F(e 0 )), since V £ W. So we need only verify
(44:E:b) of 44.7.3.

— ► — >

Consider a /3 in E(e<>), but not in V. Then 0 is also in F(e 0 ) but not in
w, hence there exists an a in W with a H $ (W is a solution for F(eo)0-
If this a belongs to E(e 0 ), then it belongs to Wn E(e 0 ) = V, i.e. we have
an a in E(e 0 ) with a H P .

If a does not belong to E(e 0 ), then it belongs to W — E(e 0 ) = D*(e 0 ),

and so it is detached. Thus a h p t a detached and in F(e 0 ). Hence there

exists by (45 :K) an a ' h P , a ' detached and in E(e o). By (45 :G) this a '
belongs to W, (E(e 0 ) sF(e 0 ), W is a solution for F(e 0 )!); hence it belongs

to Wn E(e 0 ) = V. So we have an a ' in E{e 0 ) with a ' H P ]

Thus (44:E:b) of 44.7.3. holds at any rate.

(45:N) If V is a solution for E(co ), then the W of (45:12) in (45:1)

is a solution for F(e 0 ).

Proof: W zF(e 0 ) is clear, so we must prove (44:E:a), (44:E:b) of 44.7.3.
Ad (44:E:a): Assume a H p for two a , P in W. a and (45 :D)
exclude that P be detached. So p is not in D*(e 0 ), hence it is in

w - D*(e o) = V.

STRUCTURE OF THE EXTENDED THEORY

379

Hence a 8 excludes that a too be in (the solution) V. So a is in

W - V = D*(e 0 ).

►

Consequently a is detached.

Now (45 :K) produces an a ' h 0 which is detached and in E(e o). Being

detached, a' belongs by (45 :G) to (the solution for E(e 0 )) V. As a 0

both belong to (the solution) V and a ' H 0 , this is a contradiction.

Ad (44:E:b): Consider a 0 = {0i, • • • , 0 n ) in F(e 0 ), but not in W.

Now form 0 (c) = { 01 (c), • • • , 0 n (c)} = { 0 i + t, • • * , 0 n + e) for every
€ 5 = 0 . Let c increase from 0 until one of these two things occurs for the
first time:

(45:17) 0 (e) is in E(e o), 1

(45:18) 0 (c) is detached. 2

We distinguish these two possibilities:

— >

(45:17) happens first, say for c = ci ^ 0: 0 (ci) is in E(e 0 ) y but it is not
detached.

— > — > — ►

If ci = 0 , then 0 = 0 ( 0 ) is in E(e 0 ). As 0 is not in V 5 W, there

exists an a H 0 in (the solution for E(e 0 )) V. A fortiori a in W.

Assume next ci > 0 , and 0 (ci) in V. As 0 (ci) is not detached, there
exists a (non-empty) S £ I with ^ 0 <(ei) < v(S). Besides, always

* in S

ft(ci) > 0». So 0 (ci) H 0 . And 0 (ci) is in V, hence a fortiori in W.
Assume, finally, ci > 0 and 0 (ci) not in V. As 0 (ci) is in E(eo), there

exists an a ^ 0 (ci) in (the solution for E(e 0 )) V. Since always 0 i(ci) > 0 ,-,

— ► — ► — > — ► — ► t

a h 0 (ci) implies a H 0by(45:J). And a is in V, hence a fortiori in W.

(45:18) happens first, or simultaneously with (45:17), say for c = 6 * ^ 0:
0 (cj) is still in F(e 0 ), and it is detached.

If 0 («.) is in E(e 0 ), then it is by (45 :G) in (the solution for E{e o)) V.
If 8 («j) is not in E(e 0 ), then it is in D*(e 0 ). So 8 («*) is at any rate in W.
— ► — > — >

1 1.e. the excess of 0 («) is - e 0 . For 0 (0) - 0 is in F(e 0 ), i.e. its excess b ^ e 0 ,
— >

and the excess of 0 («) increases with c,

1 I.e. ^ 0,(«) ^ v(S) for allS £ /. Each ^ 0,(«) increases with c.

% in 3

i in S

380 COMPOSITION AND DECOMPOSITION OF GAMES

This excludes «i = 0, since 0 = 0 (0) is not in W. So c* > 0.

For 0 < € < €i, p («) is not detached, so there exists a non-empty S £ I
with 2 £*(*) < v(/S). Hence there exists by continuity a non-empty

tin 8

Ssl even with £ ft(€*) rg v(S). Besides, always ft(«i) > ft, hence

tinS

P (ci) H 0 . And 0 («*) belongs to W.

Summing up: In every case there exists an a h p in W. (This a was

a , 0 («i), a , p (€j) above, respectively.) So (44:E:b) is fulfilled.

We can now give the promised proof :

Proof of (45:1): Immediate, by combining (45 :L), (45 :M), (45 :N).

45.6. Summary and Conclusions

45 . 6 . 1 . Our main results, obtained so far, can be summarized as follows:
(45:0) If

(45:0 :a) < — |r| lf

then E(e 0 ), F(e 0 ) are empty and the empty set is their only
solution.

If

(45:0:b) — |r|i ^ e 0 ^ |r| t ,

then E(eo), F(e 0 ) are not empty, both have the same solutions,
which are all not empty.

If

(45:0 :c) e 0 > |r| t ,

then E(e o), F(e 0 ) are not empty, they have no solution in common,
all their solutions are not empty.

Proof: Immediate by combining (45: A), (45:1) and (45 :H).

This result makes the critical character of the points e 0 = — |F| i, |r|*
quite clear and it further strengthens the views expressed at the end of
45.1. and following (45 :H) in 45.4.2. concerning these points: That it is here
where e 0 becomes “too small 9 ’ or “too large 99 in the sense of 44.6.1.

46 . 6 . 2 . We are also able now to prove some relations which will be useful
later (in 46.5.).

(45 :P) Let W be a non-empty solution for F(e 0 ), i.e. assume that

€o ^ — |r|i. Then

(45:P:a)

e( a ) = eo

DETERMINATION OF ALL SOLUTIONS

381

(45:P:b)

Min ^ inW «(«) = Min («„ |r|,).>

Also

(45:P:c) Max-* in ^ e ( « ) — Min-* in w «( « ) = Max (0, eo — |r|*).*

Proof: (45:P:c) follows from (45:P:a), (45:P:b) since

Co — Min (e 0 , |T|j) = Max (c 0 — e 0 , e 0 — j r|a) = Max (0, e 0 — |r|*).

We now prove (45:P:a), (45:P:b).

Write W = V u D*(e o), V a solution for E(e o), following (45:1). As

e B * — |r| )( so Vis notempty (by (45:A) or (45:0)). As we know e( a ) = eo

throughout V and e( a) < e 0 throughout D*(e 0 ).

Now for e 0 g |r| 2 , D*(e 0 ) is empty (by (45:H)), so

(45:19)

(45:20)

MaX ainW e(a) = MaX « inV e(a) =e °’

— > ►

Min-* . e( a ) = Min-* . e( a ) = e 0 .
a in W a in V

And for e 0 > |r| 2 , Z>*(e 0 ) is not empty (again by (45:H)), it is the set of all

detached a with e( a ) < e 0 . Hence by (45:B:b) in 45.2.3. these e( a) have
a minimum, |r| 2 . So we have in this case:

(45:19*) Max—. w/ e( a ) = Max- . e(a)=e 0 ,

« in W a in V

(45:20*) Min-* . e(Z) = Min-*. e(7) = |r|,.

« in W a in D*(e 0 )

(45:19), (45:19*) together give our (45:P:a), and (45:20), (45:20*) give
together our (45:P:b).

46. Determination of All Solutions in a Decomposable Game
46.1. Elementary Properties of Decompositions

46.1.1. Let us now return to the decomposition of a game T.

Let T be decomposable for J, K( = I — J) with A, H as its K- con-

stituents.

Given any extended imputation a = {an,

, a n \ for 7, we form its

J-, ^-constituents ft , y (ft = a t for i in J, = a» for i in K), and their
excesses

1 Our assertion includes the claim that these Max— . and Min— . exist.

a in W « mw

* Verbally: The maximum excess in the solution W is the maximum excess allowed in
F(e 0 ): s 0 . The minimum excess in the solution W is again e 0 , unless e 0 > |r|*, in which
case it is only |r|*. I.e. the minimum is as nearly e 0 as possible, considering that it must
never exceed |r|*.

The “width” of the interval of excesses in W ia the excess of e 0 over |rlt, if any.

382 COMPOSITION AND DECOMPOSITION OF GAMES

(46:1)

Since

Excess of a in I: e = e( a) = X a, — v(/j,

»- x

Excess of 0 in J: f = f( a) = X a< — v(J),

* in /

Excess of 7 inJf:p = ^(a) = X a< — v(-K). 1

» in K

(46:2) v(J) + v(X) = v(/)

(by (42:6 :b) in 42.3.2., or equally by (41:6) in 41.3.2. with S = J, T = K)
therefore

(46:3) e=f + g

(46 :A) We have

(46 :A:a) |rU = |A| X + |H| lf

(46 :A :b) |r | 2 = |A|* + |H| 2 .

(46:A:c) r is inessential if and only if A, H are both inessential.

Proof: Ad (46:A:a): Apply the definition (45:2) in 45.1. to T, A, H ir
turn.

(46:4) |r|i = v(J) - X v((*)),

i in I

(46:5) |A|i = v(J) - X v((i)),

* in J

(46:6) |H|i = v(K) - X v((i)).

t in K

Comparing (46:4) with the sum of (46:5) and (46:6) gives (46:A:a), owing t<
(46:2).

Ad (46:A:b): Let a , p , y be as above (before (46:1)). Then a ii
detached (in J) if

X ^ v(fl) for all R £ /.

* m R

Recalling (4t:6) in 41.3.2. we may write for this

(46:7) X “<+ H “i^ v (5)+v(T) for all SsJ, TsK.

i in S i in T

Again 0 , y are detached (in J, K ) if

(46:8) for all SsJ,

i in 8

(46:9)

X «< £ v(T) for all TsK.

tin T

1 Up to this point it was not necessary to give explicit expression to the dependenc
— ► — >

of a ' s excess e upon a . We do this now.for e as well as for /, g.

DETERMINATION OF ALL SOLUTIONS

383

Now (46:7) is equivalent to (46:8), (46:9). Indeed: (46:7) obtains by adding
(46:8) and (46:9) ; and (46:7) specializes for T = © to (46:8) and for S — @
to (46:9).

Thus a is detached, if and only if its ( J-, K - ) constituents /9 , y are both
detached. As their excesses e and /, g are correlated by (46:3), this gives
for their minima (cf. (45:B:b))

|r|t = |A|s + |H|t,

i.e. our formula (46:A:b).

Ad (46:A:c): Immediate by combining (46:A:a) or (46:A:b) with (45 :E)
as applied to r, A, H.

The quantities |r|i, |r| 2 are both quantitative measures of the essentiality
of the game r, in the sense of 45.3.1. Our above result states that both are
additive for the composition of games.

46 . 1 . 2 . Another lemma which will be useful in our further discussions:

(46 :B) If a H j8 (for r), then the set S of 30.1.1. for this domination
can be chosen with S £ J or S £K without any loss of generality. 1

Proof : Consider the set S of 30.1.1. for the domination a H (3. If
accidentally Sfi/ or S £ K, then there is nothing to prove, so we may
assume that neither S £ J nor S £ K. Consequently S = Si u T h where
Si £J, Ti £ K , and neither Si nor Ti is empty.

We have a t > ft for all i in S } i.e. for all i in Si, as well as for all i in 7\.
Finally

2 at* v(<s).

« in 8

The left hand side is clearly equal to ^ a* + ^ a», while the right hand

i in Si i in T x

side is equal to v(iSi) + v(7\) by (41:6) in 41.3.2. Thus

2) «< + X — v ^ 1 ) +

i in S i t in T x

hence at least one of

'Z «< = v ( Si )> Z ai ^

must be true.

Thus of the three conditions of domination in 30.1.1. (for a & P )
(30:4:a), (30:4:c) holds for both of Si, Ti and (30:4:b) for at least one of
them. Hence, we may replace our original S by either Si (s J) or J\(£ K).

This completes the proof.

1 I.e. this extra restriction on S does not (in this case!) modify the concept of
domination.

384 COMPOSITION AND DECOMPOSITION OF GAMES

46.2. Decomposition and Its Relation to the Solutions: First Results Concerning F(e<>)

46 . 2 . 1 . We now direct our course towards the main objective of this
part of the theory: The determination of all solutions U/ of the decomposable
game r. This will be achieved in 46.6., concluding a chain of seven lemmas.

We begin with some purely descriptive observations.

Consider a solution U/ for F(e 0 ) of r. If U/ is empty, there is nothing
more to say. Let us assume, therefore, that U/ is not empty — owing to
(45: A) (or equally to (45:0)) this is equivalent to

Co 5; — |r| t = — |AU - |H|!.

Using the notations of (46:1) in 46.1.1. we form:

(46:10)

Max-. ., /(«) = ?,

a in U/

Min- . ../(«) = *>,

« inU/

Max- . g( a ) = ^ ,
a in Uj

Min- . g( a ) =

« inU/

1 That all these quantities can be formed, i.e. that the maxima and the minima in
question exist and are assumed, can be ascertained by a simple continuity consideration.

Indeed /( a ) * ^ a x — v(«7) and g( a ) — ^ a* — v(K) are both continuous
» in J i in K

— ►

functions of a , i.e. of its components <*i, • • • , a n . The existence of their maxima and
minima is therefore a well known consequence of the continuity properties of the domain

of a — the set U /.

For the reader who is acquainted with the necessary mathematical background —
topology — we give the precise statement and its proof. (The underlying mathematical
facts are discussed e.g. by C. Carathtodory , loc. cit., footnote 1 on p. 343. Cf. there
pp. 136-140, particularly theorem 5).

U/ is a set in the n-dimensional linear space L n (Cf. 30.1.1.). In order to be sure
that every continuous function has a maximum and a minimum in U/, we must know that
U/ is bounded and closed.

Now we prove:

(*) Any solution U for F(e 0 ) [F(e 0 )l of an n-person game r is a bounded and closed
set in L n .

Proof: Boundedness: If a ■ (ai, • • • , «n| belongs to U, then every a, ^ v((i))

n

and ^ a t - v(7) ^ e 0 , hence a t £ v(I) + e 0 - ^ a/ ^ v(/) + e 0 — ^ v((t)).

»-l ;>*

So each a x is restricted to the fixed interval

v((t)) ^ o» ^ v(7) + e Q - £ v((i)),

;V»

and so these a form a bounded set.

Closedness: This is equivalent to the openness of the complement of U* That set is,

— > — ►

by (30:5:c) in 30.1.1., the set of all 0 which are dominated by any a of U. (Observe

DETERMINATION OF ALL SOLUTIONS

385

Given two a = {oi, •••,<*„}, p = {/9i, • • • , fi n ] there exists a
unique y = {yi, • • • , ?„} which has the same ./-component as a , and
the same if-component as p :

(46:11)

46 . 2 . 2 . We now prove:

y< = a ( for i in J,

y i = pi for i in K.

(46 :C)
(46:C:a)

(46:C:b)

If a , P belong to U/, then the y of (46:11) belongs to U/
if and only if

f{a) + g(p) £ e 0 .

Incidentally

«(y) =/(«) +g(P)-

Proof: Formula (46:C:b): By (46:3) in 46.1.1. e(y) = /( y ) + g( y ),
and clearly /( y ) =f(a),g(y) = g(P).

* — ►

Necessity of (46:C:a): Since U/£F(e 0 ), therefore e( y ) S to is neces-
sary and by (46:C:b) this coincides with (46:C:a).

— >

Sufficiency of (46:C:a): y is clearly an extended imputation, along

with a , P , and (46:C:a), (46:C:b) guarantee that y belongs to F(e 0 ). 1

Now assume that y is not in U/. Then there exists a 5 H y in Uo
The set S of 30.1.1. for this domination may be chosen by (46 :B) with

— ► — ► — ► >

S J or S S K . Now clearly 8 y implies, when SsJ that 8 H a ,

that we are introducing the solution character of U at this point!)

For any a denote the set of all 0 •-* a by D-+. Then the complement of U is

> a

the sum of all D->, a of U.

a

Since the sum of any number (even of infinitely many) open sets is again open, it
suffices to prove the openness of each D->, i.e. this: If 0 « , then for every 0 '

a

which is sufficiently near to 0 , we have also 0 ' h a . Now in the definition of dom-
— > — > — *

ination, 0 <-$ a by (30:4:a)-(30:4:c) in 30.1.1., 0 appears in the condition (30:4 :c) only.
And the validity of (30:4:c) is clearly not impaired by a sufficiently small change of 0< t
since (30:4 :c) is a < relation.

— * — *

(Note that the same is not true for a , because a appears in (30:4:b) also, and
(30:4:b) might be destroyed by arbitrary small changes, since (30:4:b) is a ^ relation.

— > — ►

But we needed this property for 0 , and not for a !)

1 This is the only use of (46:C:a).

386 COMPOSITION AND DECOMPOSITION OF GAMES

and when S £ K that 6 H p . As 6 , a , /3 belong to Uz, both alterna-
tives are impossible.

Hence y must belong to Uz, as asserted.

We restate (46 :C) in an obviously equivalent form:

(46 :D) Let V/ be the set of all /-constituents and W* the set of all

K - constituents of Uz.

Then Uz obtains from these V/ and W* as follows:

Uz is the set of all those y , which have a /-constituent a '
in V/ and a K - constituent 0 ' in Wx such that

(46:12)

e( a ') + e( 0 ') ^ e 0 .

i

46.3. Continuation

46.3. Recalling the definition of U/’s decomposability (for/, K) in (44 :B)
in 44.3.1., one sees with little difficulty, that it is equivalent to this:

Uz obtains from the V/, Wx of (46 :D) as outlined there, but without
the condition (46:12).

Thus (46:12) may be interpreted as expressing just to what extent Uz
is not decomposable. This is of some interest in the light of what was said
in 44.3.3. about (44:D:a) there.

One may even go a step further: The necessity of (46:12) in (46 :D) is
easy to establish. (It corresponds to (46:C:a), i.e. to the very simple first
two steps in the proof of (46 :C)). Hence (46 :D) expresses that Uz is no
further from decomposability, than unavoidable.

All this, in conjunction with (44:D:b) in 44.3.3., suggests strongly that
V/, W k ought to be solutions of A, H. With our present extensions of all
concepts it is necessary, however, to decide which F (/<>), F(g 0 ) to take;
/o being the excess we propose to use in /, and g 0 the one in K . 2 It will
appear that the <p f of 46.2.1. are these / 0 , go .

Indeed, we can prove:

(46 :E)

(46:E:a) Vj is a solution of A for F(p),

(46:E:b) Wzc is a solution of H for F(^).

It is convenient, however, to derive first another result:

1 Note that these a ', 0 ' are not the a , 0 of (46 :C) — they are their /-, if -constit-
uents as well as those of y . e( a ; ), e( 0 ') are the excesses of a 0 ' formed in /,

K. But they are equal to /( a ), g( 0 ) as well as to /( y ), g( y ). (All of this is related
to (46:C)).

* The reader will note that this is something like a question of distributing the given
excess e 0 in / between / and K.

(46 :F)

(46:F:a)

(46:F:b)

DETERMINATION OF ALL SOLUTIONS

387

9 + $ — Co,

<£ + i = e 0 .

Note that in (46 :E), as well as in (46 :F), the parts (a), (b) obtain from
each other by interchanging J, A, 9, with K, H, f. Hence it suffices to
prove in each case only one of (a), (b) — we chose (a).

— y — y

Proof of (46:F:a): Choose an a in U / for which /( a ) assumes its maxi-

— y — >

mum <p . Since necessarily e( a ) ^ e 0 , and since by definition g( a)
therefore (46:3) in 46.1.1. gives

(46:13) $5 -|- ^ 6 q .

Assume now that (46:F:a) is not true. Then (46:13) would imply
further

(46:14) $ + f < Co.

Use the above a in U/ with/( a ) = p, and chose also a & in U/ for which

g{ p ) assumes its minimum Then f( a) + g( P) = P + (by

(46:13) or (46:14)). Thus the y of (46 :C) belongs to U /, too. Again
(46 :C) together with (46:14) gives

«( y ) = /(*)+ fK £ ) = P + £ < eo,

n

i.e. 2) 7 » < v(J) + «o- Now define

<-i

S = (5i, • • • , i»l = {7i + «,*••, 7n + «),

n ^

choosing e > 0 so that ^ = v(/) + e 0 . Thus 5 belongs to F(e 0 ).

%-i

If 5 did not belong to U /, then an rj H 5 would exist in U /. By
( 45 :J) 1} H 7 , which is impossible, since ij, 7 are both in U /. Hence
5 belongs to U/. Now £ - v(J) > £ — v(J) = £ a* — v(J) }

i in J t in J * in J

i.e. /( 5 ) > /( a ) =9, contradicting the definition of 9.

Consequently (46:F:a) must be true and the proof is completed.

Proof of (46:E:a): If a ' belongs to V/, then it is the /-constituent of an

a of Uj. Hence (cf. footnote 1 on p. 386) e( a ')=/(« ) Ss 9, so that a '
belongs to P(9). Thus V/ £ P(9)-

388 COMPOSITION AND DECOMPOSITION OF GAMES

So our task is to prove (44:E:a), (44:E:b) of 44.7.3.

Ad (44:E:a): Assume, that a ' h p ' happened for two a ', 0 ' in V/.

— — > — > — ► — > —4

Then a P ' are the /-constituents of two y , 8 in U/. But a ' h 0 '

— ► —4

clearly implies 7 H 6 , which is impossible.

—4

Ad (44:E:b) : Consider an a ' in F(<p) but not in V/. Then by definition

~4 — ►

e( a ') ^ £. Use the 0 in U/ mentioned in the above proof of (46:F:a),
— ► — > -4

for which g( p ) = Let P ' be the /^-constituent of this P , so that

P ' is in Wit and e(P') = g( p ) = Thus e( a ') + e( P ') ^ $ + £ = e 0

(use (46:F:a)). Form the 7 (for /), which has the /-, /^-constituents

a 0 '. Then e(y)=e(a')+e(fi')£ e 0 i.e. 7 belongs to F(e 0 ).

7 does not belong to U/ because its /-constituent a ' does not belong

to V/. Hence, there exists a 5 H 7 in (the solution for F(e 0 )) U/.

Let S be the set of 30.1.1. for the domination 5 7 . By (46 :B) we

may assume that S £ / or S c K.

Assume first that S zK. As 7 has the same -^-constituents p ' as

P , we can conclude from 8 h 7 that 6 H p . Since both ~8 y ~p
belong to U/ f this is impossible.

Consequently SsJ, Denote the /-constituent of 8 by 5'; as V

belongs to U/, therefore 8 ' belongs to Vj. 7 has the /-constituent a '.

Hence we can conclude from 6 H 7 that 5 ' H a '.

— ► — > — ►

Thus we have the desired 8 ' from V/ with 8 ' H a

46.4. Continuation

46 . 4 . 1 . (46 :D), (46 :E) expressed the general solution U/ of T in terms
of appropriate solutions of Vj, Wk of A, H. It is natural, therefore, to
try to reverse this procedure: To start with the V/, Wx and to obtain Ui.

It must be remembered, however, that the Vj, W* of (46 :D) are not
entirely arbitrary. If we reconsider the definitions (46:10) of 46.2.1. in
this light of (46 :D), then we see that they can also be stated in this form:

Mftx ?

Ma VmW, e( ^' )= *’

( 46 : 15 )

DETERMINATION OF ALL SOLUTIONS 389

And (46 :F) expresses a relationship of these p, £ which are determined
by V/, W k — with each other and with e 0 .

46 . 4 . 2 . We will show that this is the only restraint that must be imposed
upon the V/, W x. To do this, we start with two arbitrary non-empty
solutions Vj, Wx of A, H (which need not have been obtained from any
solution U/ of T), and assert as follows:

(46 :G) Let V/ be a non-empty solution of A for F(ip) and Wx a non-

empty solution of H for F(£). Assume that ^ fulfill (46:15)
above, and also that with the <p , of (46:15)

(46:16) £ + £ = <? + # = Co-

— y - — >

For any a ' of Vy and any /3 ' of Wx with

(46:17)

e( a ') + e( & ') g e 0)

form the y (for I) which has the ^-components a
Denote the set of all these y by U/.

The U/ which are obtained in this way are precisely all
solutions of T for F(e 0 ).

Proof : All 11/ of the stated character are obtained in this way: Apply
(46 :D) to U/ forming its V/, Wx. Then all our assertions are contained in
(46 :D), (46 :E), (46 :F) together with (46:15).

All U/ obtained in this way have the stated character: Consider an U/
constructed with the help of V/, Wx as described above. We have to
prove that this U/ is a solution T for F(e 0 ).

— > — ► — > — ►

For every y of U / our (46:17) gives e( y ) = e( a ') + e{ 0 ') ^ e 0} so

— >

that y belongs to F(e 0 ). Thus U/ £ F(cq).

So our task is to prove (44:E:a), (44:E:b) of 44.7.3.

Ad (44:E:a): Assume that ij h y happened for two 17 , y in U/.

Let a 0 ' be the J-, K-constituents of y and 5 e ' the J-, X-con-

stituents of 17 from which they obtain as described above. Let S be the

set of 30.1.1. for the domination rj h y . By (46 :B) we may assume that

— > — y — ► — y

S £j or S £ K. Now S £J would cause 77 h y to imply 5 ' H a

which is impossible, since 6 ', a ' both belong to V /5 and Ss K would

— ► - — ► — > — y — y — y

cause 17 H y to imply € ' H 0 ' which is impossible, since e 0 ' both

belong to Wx.

Ad ( 44 :E:b): Assume per absurdum , the existence of a 7 in F(e 0 ) but
not in U/, such that there is no 17 of U/ with 17 h 7 . Let a 0 ' be the
«/-, ^-constituents of 7 •

390 COMPOSITION AND DECOMPOSITION OF GAMES

Assume first e( a ') ^ ip. Then a ' belongs to F(<p). Consequently
either a ' belongs to V/ or there exists a 8 ' in V/ with 5 ' h a '. In the
latter case choose an £ ' in Wx for which e( £ ') assumes its minimum value
Form the rj with the «/-, /^-constituents 8 ', € As 5 e ' belong to
V/, W k, respectively, and as e( 8 ') + e( e ') ^ £ + £ = e 0 , therefore i;

belongs to U/. Besides tj h 7 owing to 5 ' H a ' (these being their J-

— * . . — *

constituents). Thus 17 contradicts our original assumption concerning 7 .

Hence we have demonstrated, for the case under consideration, that a '
must belong to Vy.

In other words :

(46:18) Either a ' belongs to V/, or e( a ') > <p.

Observe that in the first case necessarily e( a ') ^ <p, and of course
in the second case e( a ') > p <p_. Consequently:

(46:19) At any rate e( a ') ^ f.

Interchanging J and K carries (46:18), (46:19) into these:

(46:20) Either 0 ' belongs to Wx or e(/3') >

(46:21) At any rate e{ (3 ') ^

Now if we had the second alternative of (46:18), then this gives in con-
junction with (46:21)

e( y ) = e( a ')+ e( 0 ') > <p + yp = e 0 ,

— ►

which is impossible, as y belongs F(e 0 ). The second alternative of (46:20)
is equally impossible.

Thus we have the first alternatives in both (46:18) and (46:20), i.e.
a p ' belong to V/, W*. As 7 belongs to F(e 0 ), therefore

e( a ') + e( 0 ') = e( 7 ) ^ e 0 .

— >

Consequently 7 must belong to U / — contradicting our original assumption.

46.6. The Complete Result in F(e 0 )

46 . 5 . 1 . The result (46 :G) is, in spite of its completeness, unsatisfactory
in one respect: The conditions (46:16) and (46:17) on which it depends are
altogether implicit. We will, therefore, replace them by equivalent, but
much more transparent conditions.

DETERMINATION OF ALL SOLUTIONS

391

To do this, we begin with the numbers 9 , $ which we assume to be given
first. Which solutions V/, W* of A, H for F(<p), F(\f>) can we then use in
the sense of (46 :G)?

First of all, Vj, Wit must be non-empty; application of (45 :A) or (45:0)
to A, H (instead of r) shows that this means

(46:22) *fe-|A|„ * -|H|,.

Consider next (46:15). Apply (45:P) of 45.6.1. to A, H (instead of T).
Then (45:P:a) secures the two Max-equations of (46:15), while (45:P:b)
transforms the two Min-equations of (46:15) into

(46:23) f = Min (?, |A|»), f = Min (*, |H|i).

Let us, therefore, define <p, i by (46:23).

Now we express (46:16), i.e.

(46:16) p + = + ^ = e o-

The first equation of (46:16) may also be written as

9 - v = ^ — i,

i.e. by (46:23)

(46:24) Max (0, 9 - |A| 2 ) = Max (0, - IH^). 1

46.5.2. Now two cases are possible:

Case (a): Both sides of (46:24) are zero. Then in each Max of (46:24)
the 0-term is ^ than the other term, i.e. 9 — |A|* ^ 0, £ — |H|j ^ 0, i.e.,

(46:25) 9 £ |A| t , * ^ |H|,.

Conversely: If (46:25) holds, then (46:24) becomes 0 = 0, i.e. it is auto-
matically satisfied. Now the definition (46:23) becomes

(46:26) f = 9, f = if,

and so the full condition (46:16) becomes 2
(46:27) 9 + if = e«.

(46:25) and (46:27) give also

(46:28) e 0 S |A|t + |H|* = |r|*.

Case (b): Both sides of (46:24) are not zero. Then in each Max of
(46:24) the 0-term is < than the other term — i.e. <p — |A| S > 0, ^ — |H|» > 0,
i.e.

(46:29) 9 > |A|«, if > |H|,.»

1 Cf. (45:P:c) and its proof.

* Of which we used only the first part to obtain (46:24), on which this discussion is
based.

1 Note that the important point is that (46:25), (46:29) exhaust all possibilities — i.e.
that we cannot have £ ^ |A|a, $ > |H|», or £ > |A|i» $ £ |H|t. This is, of course, due to
the equation (46:24), which forces that both sides vanish or neither.

The meaning of this will appear in the lemmas which follow.

392 COMPOSITION AND DECOMPOSITION OF GAMES

Conversely: If (46:29) holds, then (46:24) becomes — |A| 2 = ^ — |H|j,
i.e. it is not automatically satisfied. We can express (46:24) by writing

(46:30) 9 — |A| a + w, # = |H|j + «,

and then (46:29) becomes simply

(46:31) a > 0.

Now the definition (46:23) becomes

(46:32) <fi = |A|j, $ = |H| S ,

and so the full condition (46:16)* becomes

|A|, + |H|j + w = eo,
i.e.

(46:33) e„ = |r|* + w.

(46:31) and (46:33) give also

(46:34) e 0 > |T|*.

46 . 6 . 3 . Summing up :

(46:H) The conditions (46:16), (46:17) of (46:G) amount to this:

One of the two following cases must hold :

Case (a): (1) -|r|, £ e, £ |r|«,

together with

(2) -|A|i £ 9 £ |A|t,

[(3) -|H|iS*S|H|„

and

(4) 9 + i = Co.

Case (b): (1) e 0 > |r| 3 ,

together with

(2) 9 > |A|t,

(3) ^ > |H| S ,

and

(4) e 0 - |r|* - 9 - |A|t = * - |H|,.»

Proof: Case (a): We knew all along, that eo ^ — |r|i and 9 2: — |A|j,
# 2 — |H|i. The other conditions coincide with (46:28), (46:25), (46:27)
which contain the complete description of this case.

Case (b): These conditions coincide with (46:34), (46:29), (46:30),
(46:33) which contain the complete description of the case (after elimination
of <■> which subsumes (46:31) under (l)-(3)).

1 Cf. footnote 2 on p. 391.

1 The reader will note that while (l)-(3) for (a) and for (b) show a strong analogy,
the final condition (4) is entirely different for (a) and for (b). Nevertheless, all this was
obtained by the rigorous discussion of one consistent theory I
More will be said about this later.

DETERMINATION OF ALL SOLUTIONS

393

46.6. The Complete Result in E(e 0 )

46 . 6 . (46 :G) and (46 :H) characterize the solutions of T for F(e 0 ) in a
complete and explicit way. It is now apparent, too, that the cases (a), (b)
of (46 :H) coincide with (45:0:b), (45:0:c) in 45.6.1.: Indeed (a), (b) of
(46 :H) are distinguished by their conditions (1), and these are precisely
(45:0:b), (45:0:c).

We now combine the results of (46:G), (46:H) with those of (45:1),
(45:0). This will give us a comprehensive picture of the situation, utilizing
all our information.

(46:1) If

(46:I:a) (1) e 0 < -|r| lf

then the empty set is the only solution of T, for E(e 0 ) as well as
for F(e 0 ).

If

(46:I:b) (1) — |r| t ^ 6 0 ^ |r| t ,

then T has the same solutions 0/ for E(e 0 ) and for F(e 0 ). These
0/ are precisely those sets, which obtain in the following manner:
Choose any two £, # so that

(2) — |AU ^ 9 ^ |A|.,

(3) — JH|i |H| t ,

and

(4) ? + £ = e 0 .

Choose any two solutions V/, W* of A, H for E((p), E{ty).

Then 0/ is the composition of V/ and W* in the sense of
44.7.4.

If

(46:I:c) (1) *o > |r| 2|

then T does not have the same solutions 0/ for E(e 0 ) and U / for
F(e 0 ). These 0/ and U/ are precisely those sets which obtain
in the following manner: Form the two numbers p, ^ with

(2) 9 > |Ak

(3) ^ > |H| 2 ,
which are defined by

(4) e 0 ~ |r| 2 = <P - |A| 2 = $ - |H| 2 .

Choose any two solutions V/, W* of A, H for 2?($), E$).
Then 0/ is the sum of the following sets: The composition of

\fj and of the set of all detached 0 ' (in K ) with e{ fi ') = |H|*; the

composition of the set of all detached a ' (in J) with e( a ') = |A|*

394 COMPOSITION AND DECOMPOSITION OF GAMES

and of WxJ the composition of the set of all detached a ' (in J)

— > — >

with e( a ') = <p and of the set of all detached 0 ' (in K) with

e( 0 ') = 4/y taking all pairs <e>, ^ with

(5) | A | 2 < <P < P, |H|, < t < ft

and

(6) <p + yp = e 0 .

U/ obtains by the same process, only replacing the condition (6)
by

(7) <p + ^ ^ €q.

Proof : Ad (46:I:a): This coincides with (45:0:a).

Ad (46:I:b): This is a restatement of case (a) in (46 :H) except for the
following modifications:

First: The identification of the E and F solutions for F, A, H. This is
justified by applying (45:0:b) to T, A, H which is legitimate by (1), (2), (3)
of (46:I:b).

Second: The way in which we formed 0/ = 11/ from Vj = V/, W* = W k
which differed from the one described in (46 :H) insofar as we omitted
the condition (46:17). This is justified by observing that (46:17) is
automatically fulfilled: V/ = Vj s E(<p), Wk = W* £ #(ft, hence for

a ' in V/ and 0 ' in W* always e( a ') = <f>, e( 0 ') = ^ and so by (4)
e( a ') + e( 0 ') = 6o.

Ad (46:I:c): This is a restatement of case (b) in (46:H), except for this
modification:

We consider both E and F solutions for T (not only F solutions as in
(46:H)), and use only E solutions for A, H (not F solutions as in (46:H)).
The way in which the former 0/, U/ of T are formed from the latter (Vj of
A, We of H) is accordingly different from the one described in (46 :H).

In order to remove these differences, one has to proceed as follows:
Apply (45:1) and (45:0:c) to T, A, H which is legitimate by (1), (2), (3)
of (46:I :c). Then substitute the defining for the defined in (46 :H). If these
manipulations are carried out on (46 :H) (in the present case (46:I:c)), then
precisely our above formulation results. 1

46.7. Graphical Representation of a Part of the Result

46 . 7 . The results of (46:1) may seem complicated, but they are actually
only the precise expression of several simple qualitative principles. The
reason for going through the intricacies of the preceding mathematical
derivation was, of course, that these principles are riot at all obvious, and
that this is the way to discover and to prove them. On the other hand our
result can be illustrated by a simple graphical representation.

1 If the reader carries this out, he will see that this transformation, although somewhat
cumbersome, presents absolutely no difficulty.

DETERMINATION OF ALL SOLUTIONS

395

We begin with a more formalistic remark.

A look at the three cases (46:I:a)-(46:I:c) discloses this: While nothing
more can be said about (46 :1 :a), the two other cases (46:I:b), (46:I:c)
have some common features. Indeed, in both instances the desired solu-
tions 0/, U/ of T are obtained with the help of two numbers ^ and certain
corresponding solutions \f J} W* of A, H. The quantitative elements of the
representation of 0/, U/ are the numbers £, As was pointed out in foot-
note 2 on p. 386, they represent something like a distribution of the given
excess e 0 in I between J and K.

4

^ are characterized in the cases (46 :1 :b) and (46:I:c) by their respec-
tive conditions (2)-(4). Let us compare these conditions for (46:I:b) and
for (46:I:c).

They have this common feature : They force the excesses £, # to belong
to the same case of A, H as the one to which the excess e 0 belongs for I\

They differ, however, very essentially in this respect: In (46:I:b) they
impose only one equation upon £, $ while in (46 :1 :c) they impose two equa-
tions. 1 Of course, the inequalities too, may degenerate occasionally to
equations (cf. (46:J) in 46.8.3.), but the general situation is as indicated.

The connections between e 0 and p, # are represented graphically by
Fig. 69.

1 (2), (3) are inequalities in both cases. (4) stands for one equation in (46:1 :b) and
for two equations in (46:1 :c).

396 COMPOSITION AND DECOMPOSITION OF GAMES

This figure shows the p, ^-plane and under it the 60-line. On the latter
the points — |r|i, |F| S mark the division into the three zones corresponding
to cases (46:I:a)-(46:I:c). The p, ^-domain which belongs to case (46:I:b)
covers the shaded rectangle marked (b) in the p, ^-plane; the p, ^-domain,
which belongs to case (46 :1 :c) covers the line marked (c) in the <p, ^-plane.

Given any £, ^-point, following the line leads to its eo value —

thus b } V yield a, a', respectively. Given any e 0 -value the reverse process
discloses all its <p y ^-points, thus a produces an entire interval at b f while
a ' yields the unique point 6'. 1

46.8. Interpretation : The Normal Zone. Heredity of Various Properties

46 . 8 . 1 . Figure 69 calls for further comments, which are conducive to a
fuller understanding of (46:1).

First: There have been repeated indications (for the last time in the com-
ment following (45:0)), that the cases (46:I:a) and (46:I:c), i.e. e 0 < — |r|i,
and 60 > |r|j, respectively — are the “too small” or “too large” values of e 0
in the sense of 44.6.1.; i.e., that case (46:I:b), — \T\i ^ e 0 ^ |r| 2 , is in some
way the normal zone. Now our picture shows that when the excess eo
of T lies in the normal zone, then the corresponding excesses <p, $ of A,
H lie also within their respective normal zones. 2 * 4 In other words:

The normal behavior (position of the excess in (46 :1 :b)) is hereditary from
T to A, H.

Second: In the case (46:I:b) — the normal zone — # are not completely
determined by e 0 , as we repeatedly saw before. In case (46:I:c), on the
other hand, they are. This is pictured by the fact that the former domain
is the rectangle (b) in the p, plane, while the latter domain is only a line (c).

It is worth noting, however, that at the two ends of the case (46:I:b) —
for 60 = — |T|i, |r|* — the interval available for p, ^ is constricted to a point.*
Thus the transition from the variable $ of (46 :1 :b) to the fixed ones of
(46:I:c) is continuous.

Third: Our first remark stated that normal behavior (i.e., that the position
of the excess corresponds to (46:I:b)) is hereditary from T to A, H. It is
remarkable that, in general, no such heredity holds for the vanishing of the
excess, i.e. that e 0 = 0 4 does not in general imply <p = 0, ^ = 0. It is pre-
cisely the vanishing of the excess which specializes our present theory (of
44.7.) to the older form (of 42.4.1. which, as we know, is equivalent to the
original one of 30.1.1.). We will examine the variability of p, # when e 0 = 0
more closely in the last (sixth) remark. Before we do that, however, we give
our attention to the connection between our present theory and the older
form.

1 We leave to the reader the simple verification that the geometrical arrangements of
Fig. 69. express, indeed, the condition of (46:I:b), (46:1 :c).

* I.e.-|r|i £ e 0 £ |r|i implies -|A|i £ £ U|j, -\H |i £ ? £ \H\%, cf. (46:I:b).

1 This is one case of degeneration, alluded to ftt the end of 46.7.

4 Of course, e 0 ■■ Olies in the normal case (46:I:b): — \T\i £ 0 £ Irl*.

DETERMINATION OF ALL SOLUTIONS

397

Fourth: It is now evident that the present, wider form of the theory
must of necessity receive consideration, even if our primary interest is
with the original form alone. Indeed: in order to find the solutions of a
decomposable game T in the original sense (for eo = 0), we need solutions
for the constituent games A, H in the wider, new sense (for $ which may
not be zero).

This gives the remarks of 44.6.2. a more precise meaning: It is now
specifically apparent how the passage from the old theory to the new one
becomes necessary when the game (A or H) is looked upon as non-isolated.
The exact formulation of this idea will come in 46.10.

46 . 8 . 2 . Fifth: We can now justify the final statements about (44 :D)

in 44.3.2. and (44:D:a), (44:D:b) in 44.3.3. (46:I:b) shows that (44 :D)

is true in the case (44:1 :b), if we relinquish the old theory for the new one;
(44:I:c) shows that (44 :D) is not true in the case (44:I:c) even at that
price. Thus the desire to secure the validity of the plausible scheme of
(44 :D) motivates the passage to the new theory as well as the restriction
to case (44:I:b) — the normal case.

If we insist interpreting (44 :D), (44:D:a), (44:D:b) by the old theory,
then (44 :D), (44:D:a) fail, 1 while the conditional statement of (44:D:b)
remains true. 2

46 . 8 . 3 . Sixth: We saw that e 0 = 0 does not in general imply p = 0,
^ = 0. What does this “in general” mean?

£, $ are subject to the conditions (2)-(4) of (46:I:b). As e 0 = 0, so
(4) means that # = — ? and permits us to express the remaining (2), (3) in
terms of $ alone. They become this:

(46:35)

|AU

|H|i

Now apply (45 :E) to A, H. Then we see:

If A, H are both essential, then the lower limits of (46:35) are < 0 and
the upper limits are > 0, so £ can really be ^ 0. If either A or H is inessen-
tial, then (46:35) implies <p = 0 and hence £ = 0.

We state this explicitly:

(46 :J) e 0 = 0 implies p = 0, # = 0, i.e. (44 :D) of 44.3.2. holds even

in the sense of old theory if and only if either A or H is inessential.

46.9. Dummies

46 . 9 . 1 . We can now dispose of the narrower type of decomposition,
described in footnote 1 on p. 340 — the addition of “dummies” to the game.

Consider the game A of the players 1 ',•••,&'.* “ Inflate ” it by adding
to it a series of “dummies” K ; i.e. compose A with an inessential game H
of the players 1", • • • , l ". Then the composite game is V.

1 As we may have e 0 “ 0, <p ^ 0, # & 0. Then the decomposability requirement
(44:B:a) of 44.3.1. is violated, as stated in 44.3.3.

* Representing the special case Co * 0, £ *» 0, ^ 0.

* It is now convenient to reintroduce the notations of 41.3.1. for the players.

398 COMPOSITION AND DECOMPOSITION OF GAMES

We will use the old theory for all these games. By (31:1) in 31.2.1.
there exists precisely one imputation for the inessential game H — say

= M", • • • , 7?"}. 1 By (31:0) or (31 :P) in 31.2.3. H possesses a

unique solution: The one element set (y ° K ).

Now by (46 :J) and (46:I:b) the general solution of T obtains by com-
posing the general solution of A with the general solution of H — and the
latter one is unique!

In other words:

“Inflate” every imputation = {j8r, • • • , 0*'} of J (i.e. A) to an
— ► — ►

imputation a / of I (i.e. r) by composing it with y ° K , i.e. by adding to it the

components 7 ?-, • • • , 7 «/ = [Pv, * * * , Pv, 7 ?", • * * > 7*°"}. Then
this process of “inflation” — i.e. of composition — produces the general solu-
tion of T from the general solution of A.

This result can be summed up by saying that the “inflation” of a game
by the addition of “dummies” does not affect its solution essentially — it is
only necessary to add to every imputation components representing the
“dummies,” and the values of these components are the plausible ones:
What each “dummy” would obtain in the inessential game H, which
describes their relationship to each other.

46 . 9 . 2 . We conclude by adding that (46 :J) states that if and only if
the composition is not of the special type discussed above, the old theory
ceases to have the simple properties of the new one, and its hereditary
character fails, as indicated in the third remark of 46.8.1.

46.10. Imbedding of a Game

46 . 10 . 1 . In the fourth remark of 46.8.1. we reaffirmed the indications of
44.6.2., according to which the passage from the old theory to the new one
becomes necessary when the game is looked upon as non-isolated. We will
now give this idea its final and exact expression.

It is more convenient this time to denote the game under consideration
by A and the set of its players by J. It ought to be understood that this A
is perfectly general — no decomposability of A is assumed.

We begin by introducing the concepts which are needed to treat a given
game A as a non-isolated occurrence: This amounts to imbedding it without
modifying it, into a wider setup, which it is convenient to describe as
another game r. We define accordingly: A is imbedded into T, or T is an
imbedding of A, if r is the composition of A with another game H. 2 In
other words, A is imbedded in all those games of which it is a constituent.*

1 Recall the notations of 44.2.

* The game H and the set of its players K are perfectly arbitrary, except that K and J
must be disjunct.

* Since a constituent of a constituent is itself a constituent (recall the appropriate
definitions, in particular (43 :D) in 43.3.1.), an imbedding of an imbedding is again an
imbedding. In other words; Imbedding is a transitive relation. This relieves us from
considering any indirect relationships based upon it.

DETERMINATION OF ALL SOLUTIONS

399

46. 10.2. Let us now investigate the solutions of A viewing A as a non-
isolated occurrence. In the light of the above, this amounts to enumerating
all solutions of all imbeddings T of A, and interpreting them, as far as A is
concerned. The last operation must be the taking of the ./-constituent
in the sense of 44.7.4. We know from the fifth remark in 46.8.2. that this
is only feasible, if we consider no solutions from outside the normal zone (b).

One might hesitate whether the solutions of T should be taken in the
sense of the old or the new theory. The former may appear to be more
justified on the standpoint of 44.6.2.: The outside influences upon the game
having been accounted for by the passage from A to H, there is no longer
any excuse for going outside the old theory. 1 It happens, however, that
we need not settle this point at all, because the result for A will be the
same, irrespectively of which theory we use for T. But if we use the new
theory for T, we must restrict ourselves to the case (46:1 :b), as discussed
above.

Thus the question presents itself ultimately in this form:

(46 :K) Consider all imbeddings T of A, and all solutions of these T:

(a) in the sense of the old theory, i.e. for E{ 0),

(b) in the sense of the new theory in the normal zone, i.e. for
any E(e Q ) of (46:I:b).

Which are the ./-constituents of the solutions?

46.10.3. The answer is very simple:

(46 :L) The ./-constituents (of the T solutions) referred to in (46 :K)

are precisely the following sets: All solutions for A in the normal
zone, i.e. for any £(v>) of (46:I:b). This is true for both (a) and
(b) of (46 :K).

Proof: e 0 = 0 belongs to case (46:I:b) (cf. footnote 4 on p. 396), hence
(a) is narrower than (b). Therefore, we need only show that all the sets
obtained from (b) are among the ones described above, and that all these
sets can even be obtained with the help of (a).

The first assertion is only that of the hereditary character of the normal
zone (b).

The second assertion follows from (46:I:b), if we can do this: Given a <p
with — |A| i ^ ^ | A 1 2 , find a game H and £ with — |H|i ^ g |H|*, such

that # = 0 and that H possesses solutions for Ety). Now such an
H exists, and it can even be chosen as a three-person game.

Indeed: Let H be the essential three-person game with general y > 0.
Then by (45 2) in 45. 1 . \R\ X = 3y and by (45:9) in 45.3.3. |H|j = ^|H|i = fy.
We have required ^ = — <p and what we know now amounts to

— 37 ^ ^ g $7.

1 Besides, the transitivity pointed out in footnote 3 on p. 398, shows that any further
imbedding of T can be regarded directly as an imbedding of A.

400 COMPOSITION AND DECOMPOSITION OF GAMES

This can clearly be met by choosing y sufficiently great. Then we also need a
solution of H for E($). The existence of such a solution (for — 3y ^ f ;§ |y)
will be shown in 47.

46 . 10 . 4 . To this result two more remarks should be added:

First: If we wanted to handle the process of imbedding in such a manner
that the old theory remains hereditary, we would have to see to this: The
composition of T from A and H has to be such that e 0 = 0 implies <p = 0
(and hence £ = 0). By (46 :T) this means that either A or H are inessen-
tial. The latter means (cf. eod), that only “dummies” are added to A.

Summing up:

(46 :M) The old theory remains hereditary if and only if either the

original game A is inessential, or the imbedding is restricted to
the addition of “dummies” to A.

Second: It was suggested already in 44.6.2. to treat the outside source,
which creates the excesses and paves the way for the transition of our old
theory to the new one, as another player.

Our above result (44 :L) justifies a slightly modified view: The outside
source of 44.6.2. is the game H which is added to A — or rather the set K of
its players.

Now we have seen that the game H must be essential, in order to achieve
the desired result. Furthermore we know that an essential game must have
n ^ 3 participants, and the proof of (44 :L) showed that a suitable H with
n = 3 participants does indeed exist.

So we see:

(46:N) The outside source of 44.6.2. can be regarded as a group of

new players — but not as one player. Indeed, the minimum
effective number of members of this group is 3.

46 . 10 . 6 . The foregoing considerations have justified our passage from
the old theory to the new one (within the normal zone (b)) and clarified
the nature of this transition. We see now that the “common sense”
surmise of 44.3. fails to hold in the old theory, but that it is true in precisely
that new domain to which we changed. This rounds out the theory in a
satisfactory manner.

The leading principle of the discussions of 44.4.3.-46.10.4. was this:
The game under consideration was originally viewed as an isolated occur-
rence, but then removed from this isolation and imbedded, without modifica-
tion, in all possible ways into a greater game. This order of ideas is not
alien to the natural sciences, particularly to mechanics. The first stand-
point corresponds to the analysis of the so-called closed systems, the second
to their imbedding, without interaction, into all possible greater closed
systems.

DETERMINATION OF ALL SOLUTIONS

401

The methodical importance of this procedure has been variously empha-
sized in the modern literature on theoretical physics, particularly in the
analysis of the structure of Quantum Mechanics. It is remarkable that it
could be made use of so essentially in our present investigation.

46.11. Significance of the Normal Zone

46.11.1. The result (46:I:b) defines for every solution of the composite
game T in the normal zone — i.e. a fortiori for every solution in the sense
of the old theory — numbers p, This and the immediate properties of
p, ^ in connection with the solution, appear to be so fundamental, as to
deserve a fuller non-mathematical exposition.

We are considering two games A, H played by two disjunct sets of players
J and K. The rules of these games stipulate absolutely no physical con-
nection between them. We view them nevertheless as one game T but
this game, of course, is composite, with the two isolated constituents A, H.

Let us now find all solutions of the entire arrangement, i.e. of the com-
posite game T. Since it is not desired to consider anything outside of T, we
adhere to the original theory of 30.1.1. and 42.4. 1. 1 Then we have shown
that any such solution U / determines a number <p 2 with the following prop-
erty: For every imputation a of U / the players of A (i.e. in J ) obtain
together the amount and the players of H (i.e. in K) obtain together the
amount —<?>. Thus the principle of organization embodied in U / must
stipulate (among other things) that the players of H transfer under all
conditions the amount £ to the players of A.

The remainder of the characterization of U; — i.e. of the principle of
organization or standard of behavior embodied in it is this:

First: The players of A, in their relationship with each other, must be
regulated by a standard of behavior which is stable, provided that the
transfer of from the other group is placed beyond dispute . 3

Second: The players of H, in their relationship with each other, must
be regulated by a standard of behavior which is stable, provided that the
transfer of to the other group is placed beyond dispute . 4

Third: The octroyed transfer a must lie between the limits (46:35)
of 46.8.3.

(46:35)

jp ^

46 . 11 . 2 . The meaning of these rules is clearly that any solution, i.e.
any stable social order of r is based upon payment of a definite tribute by
one of the two groups to the other. The amount of this tribute is an
integral part of the solution. The possible amounts, i.e. those which can

1 I.e. e. - 0.

* Since £ + ^ - e 0 - 0, we do not introduce f - -$•

1 I.e. that the /-constituent V/ of U/ is a solution of A for E($).

* I.e. that the ^-constituent W* of U/ is a solution of H for

402 COMPOSITION AND DECOMPOSITION OF GAMES

occur in solutions, are strictly determined by (46:35) above. This condi-
tion shows in particular:

First: The tribute zero, i.e. the absence of a tribute is always among the
possibilities.

Second: The tribute zero is the only possible one if and only if one of
the two games A, H is inessential (cf. the sixth remark in 46.8.3.).

Third: In all other cases both positive and negative tributes are possible
— i.e. both the players of A and the players of H may be the tribute paying
group.

The limits of (46:35) are set by both games A, H, i.e., by the objective
physical possibilities of both groups. 1 These limits express that each
group has a minimum below which no form of social organization can depress
it: — |A|i, — |H| i; and, each group has a maximum, above which it cannot
raise itself under any form of social organization: |A|t, |H| 2 .

Thus, for a particular physical background, i.e. a game, say A, the two
numbers |A|i, |A|* can be interpreted this way: — |A|i is the worst that will
be endured under any conditions and |A|* is the maximum claim which
may find outside acceptance under any conditions. 2

The results (45 :E) and (45 :F) of 45.3. 1.-2. now acquire a new signifi-
cance: According to these the two numbers can only vanish together (when
A is inessential) and their ratio always lies between definite limits.

46.12. First Occurrence of the Phenomenon of Transfer: n - 6

46 . 12 . We have seen repeatedly (thus in (46 :J) in 46.8.3. and in the
second and third remarks in 46.11.2.) that the characteristic new element
of the theory of a composite game T manifests itself only when both con-
stituents A, H are essential. This is the occurrence of e 0 = 0, but

9 = — ^ ^ 0 ,

i.e. a non-zero tribute in the sense of 46.11.

Now we know that in order to be essential a game must have ^ 3
players. If this is to be true for both A, H, then the composite game T
must have ^ 6 players.

Six players are actually enough as the following consideration shows:
Let A, H both be the essential three-person games with 7 = 1. Then
|A| ! = |H| ! = 3, (A|j = |H|i = f. (Cf. in 46.10.3.). Hence for -* £ 9 £ f,
both p and ^ ^ lie between —3 and i. This implies, as will be shown

in 47., the existence of solutions V/, W* of A, H for E (<?>), E$). Their com-

1 But where the actual amount £ lies between those limits, is not determined by those
objective data, but by the solution, i.e. the standard of behavior which happens to be
generally accepted.

* It must be recalled that all this takes the value of the coalition of all players of A,
v(/), as zero; i.e., we are discussing the losses which are purely attributable to lack of
co-operation among the group, and unfavorable general social organizations — and gains,
which are purely attributable to lack of co-operation in outside groups and favorable
general social organizations.

THREE-PERSON GAME

403

position U/ is then a solution of the composite game T with the given p.
Since was only restricted by — $ ^ <p ^ -f , we can choose it non-zero.

Thus we have demonstrated:

(46:0) n = 6 is the smallest number of players for which the char-

acteristic new element of our theory of composite games (the
possibility of e 0 = 0 with p = — # ^ 0, cf. above) can be
observed in a suitable game.

We have repeatedly expressed the belief that an increase in the number
of players need not only cause a more involved operation of the concepts
which occurred for smaller numbers, but that it also may originate quali-
tatively new phenomena. Specifically such occurrences were observed
as the number of players successively increased to 2,3,4. It is, therefore,
of interest that the same happens now as the number of players reaches six. 1

47. The Essential Three-person Game in the New Theory

47.1. Need for This Discussion

47.1. It remains for us to discuss the solutions of the essential three-
person game, according to the new theory.

This is necessary, since we have already made use of the existence of
these solutions in 46.10. and 46.12., but the discussion possesses also an
interest of its own. In view of the interpretation which we were induced
to put on these solutions in 46.12. and also of their central role in the
theory of decomposition, 2 it seems desirable to acquire a detailed knowledge
of their structure. Furthermore, a familiarity with these details will lead
to other interpretations of some significance. (Cf. 47.8. and 47.9.) Finally,
we shall find that the principles used in determining the solutions in ques-
tion are of wider applicability. (Cf. 60.3.2., 60.3.3.)

47.2. Preparatory Considerations

47.2.1. We consider the essential three-person game, to be denoted by
T, in the normalization 7 = 1. Thus \T\ { = 3, |r| 2 = f (cf. 46.12.). We
wish to determine the solutions of this T for E(e 0 ).* In the applications,
referred to above we needed only the normal zone — 3 ^ e 0 ^ $ but we pre-
fer to discuss now all e 0 .

This discussion will be carried out with the graphical method, which
we used in treating the old theory in 32. We will, therefore, follow the
scheme of 32. in several respects.

1 For some other qualitatively new phenomena which emerge only when there are six
players, cf. 53.2.

* This is the only problem of absolutely general character, of which we have a complete
solution at present!

* We are writing T, e 0 although the applications employed the notations A, ? and H,

X" -£).

Of course, the present r has nothing to do with the decomposable r considered
before.

404 COMPOSITION AND DECOMPOSITION OF GAMES

The characteristic function is the same as in 32.1.1.:

/ 0 10

(47:1) v(S) = < } when S has H elements.

J 1 J Z

V o \3

An (extended) imputation is a vector

a = {«i, a 2 , a*},

whose three components must fulfill (44:13) in 44.7.2., which becomes now:

(47:2) ai ^ -1, a 2 ^ -1, a 8 ^ -1.

Besides, in E(e 0 ) the excess must be e 0 , according to (44:11*) in 44.7.2.
and this is now

(47:3) «i + «2 + « 8 = eo. 1

47.2.2. We wish to represent these a by the graphical device of 32.1.2.
But that procedure pictures only number triplets of sum zero. Therefore
we define

(A *7 .A \ l ^0 2 CO • c 0

(47:4) a> = a 1 = a 2 — g-* a 8 = a 3 — g-*

Then (47:2), (47:3) become

(47:2*) a 1 ^ - (l + a 2 ^ - (l + a» £ - (l +

(47:3*) a 1 + a 2 + a 3 = 0. 2

Now the representation of 32.1.2. becomes applicable, we need only
replace a\, a 2 , a z by a 1 , a 2 , a s . With this reservation, Figure 52 can be
used.

— >

For these reasons, we form for every vector a = («i, a 2 , a z ] of E(e 0 )
not only its components in the ordinary sense but also its quasi-components
in the sense of (47:4): a 1 , a 2 , a 8 ; and with the help of the quasi-components,
we utilize the graphical representation of Figure 52.

So this plane representation expresses precisely the condition (47:3*).
The remaining condition (47:2*) is therefore equivalent to a restriction

imposed upon the point a , within the plane of Figure 52. This restriction

l The reader should compare (47:1)- (47:3) with (32:1)- (32:3) of 32.1.1. — the sole
difference lies in (47:3).

1 Comparing these (47:2*), (47:3*) with (32:2), (32:3) of 32.1.1., it appears that
(47:3*) and (32:3) coincide, and that (47:2*) and (32:2) differ only by the factor of
e 0

proportionality 1 + — •

THREE-PERSON GAME

406

obtains in the same way as the similar one in 32.1.2.: a must lie within
the triangle formed by the three lines a 1 = - ^1 + a 2 = — ^1 +

a* = — (l + This is precisely the situation of Fig. 53., except for the

proportionality factor 1 + and it is represented on Figure 70. The

— ►

shaded area to be called the fundamental triangle, represents the a which
fulfill (47:2*), (47:3*), i.e. those of E(e 0 ).

Figure 70.

47.2.3. We express the relationship of domination in this graphical
representation. As we are using the new theory, the considerations of 31.1.

# -4 — — >

concerning the set S of 30.1.1. for a domination a 0 — i.e. concerning

its certainly necessary or certainly unnecessary character — no longer apply.
So we discuss S de novo .

It is still true, that 8 cannot be a one- or a three-element set. In the first
case S = (i), so by 30.1.1. a» ^ v((i)) = —1, on > ft, hence ft < — 1,
contradicting ft ^ —1 by (47:2). In the second case S = (1,2,3), so by
30.1.1. ai > ft, a% > ft, as > ft, hence ai + a* + as > ft + ft + ft,
contradicting ai + a 2 + a 8 = ft + ft + ft = e 0 by (47:3).

£o

1 Cf . footnote 2 on p. 404. Here We assume, of course, that 1 + ^ ^ 0, i.e.

«o ^ —3 ** — |r|i.

If 1 + < 0, i.e. 6 0 < —3 - — |r|i, then the conditions of (47:2*), (47:3) conflict, and

indeed we know from (45 :A) that E(e Q ) is empty in this case.

406 COMPOSITION AND DECOMPOSITION OF GAMES

Thus S must be a two-element set, S = (i, j). 1 Then domination
means that a, + a, ^ v((i, j)) = 1 and > ft, a, > ft, i.e. that

a * + ^ 1

260

and a* > ft, a*' > ft. By (47 :3*) the first condition may be written

~ 0 - t)*

We restate this: Domination

means that

— > — >

a H 0

/ either a 1 > ft,

a 2 > ft and

a 3 ^ ^

1 - f )•

(47:5) (or a 1 > ft,

a 8 > ft and

a 2 ^ ^

‘ - T>

| or a 2 > ft,

a 8 > ft and

IIV

1

1 - t}*

47.3. The Six Cases of the Discussion. Cases (I)-(III)

47 . 3 . 1 . After these preparations we can proceed to discuss the solutions
V of T for E(e 0 ), for all values of e 0 .

It will be found convenient to distinguish six cases. Of these Case (I)
corresponds to (45:0:a), Cases (II)-(IV) and one point of (V) to (45:0:b)
(the normal zone), and Cases (V) and (VI) (without that point) to (45:0:c)
(all in 45.6.1.).

47 . 3 . 2 . Case (I): e 0 < -3. In this case 1 + |° < 0, so (47:2*), (47:3*)
conflict and E(e 0 ) is empty (cf. footnote 1 on p. 405) so V must be empty too.

Case (II): e 0 = —3. In this case 1 + ^ = 0, so (47:2*), (47:3*)

Pt\

imply a 1 = a 2 = a* = 0, i.e. aj = oj = a» = -x = —1,

a

« = (-1, -l, -i).

So E(e o) is a one-element set, and V must be = E(e o) by the same argument
as in the proof of (31:0) in 31.2.3. Thus the conditions are very similar
to those encountered in an inessential game, cf. loc. cit.

1 i f jy k a permutation of 1,2,3.

•This differs from the corresponding (32:4) in 32.1.3. only by the extra condition

at the end of each line.

THREE-PERSON GAME

407

Case (III): — 3 < e 0 ^ 0. In this case 1 + | # > 0, so we can use

Figure 70. Also 1 + 1 so the extra conditions of (47 :5) in

47.2.3. are automatically fulfilled throughout the fundamental triangle.
So (47 :5) coincides with (32:4) in 32.1.3. (cf. footnote 2 on p. 406). Conse-
quently the entire discussion of 32.1.3.-32.2.3. applies again, if the pro-

portionality factor 1 + ~ is inserted.

u

Thus we obtain the solutions of E(e 0 ) in this case simply by taking those

described in 32.2.3., multiplying each component by 1 + and adding

<5

^ (to pass from a* to a t ).

47.4. Case (IV) : First Part

47 . 4 . 1 . Case (IV): 0 < e 0 < | In this case 0 < 1 - ^ < 1 +
Consequently the lines

(which bound the extra conditions of (47:5)) are situated with respect to
the fundamental triangle of Figure 70 as indicated on Figure 71. They
subdivide the fundamental triangle into seven areas, each of which can be
characterized by stating which two-element sets S are effective in it in the
sense of (47:5). The list is given below Figure 71. Now we can draw
the analogue of Figure 54, indicating for each point of the fundamental
triangle the shaded areas 1 which it dominates. This is done in Figure 72
according to (47:5). It is necessary to treat each one of the seven areas of
Figure 71 separately, and every shaded area of Figure 72 must be continued
across the entire fundamental triangle.

It is clear from Figure 72, that no point of the area © can be dominated
by a point outside that area. 2 Hence the condition (44:E:c) of 44.7.3.,
which characterizes the solution V for E(e 0 ), i.e. for the entire fundamental
triangle, must also hold for the part of V in © when taken for © (in place
of the entire fundamental triangle, i.e. E(e 0 )). But ® is a triangle like the
fundamental triangle of Figure 53, except for the proportionality factor

1 — Comparison of Figure 54 with ® in Figure 72 shows that the

conditions of domination are the same.

1 Excluding their boundaries.

1 Including its boundary.

8 Note that 1 — > 0.

408 COMPOSITION AND DECOMPOSITION OF GAMES

47 * 4 . 2 . Consequently the entire discussion of 32.1.3.-32.2.3. applies

2eo

to the part of V in ®, if the proportionality factor 1 is inserted.

Area:

Effective two-element
sets S :

®

(1,2), (1,3), (2,3)

(1,2), (1,3)

(1, 2), (2, 3)

®

(1, 3), (2, 3)

®

(2, 3)

®

0, 3)

®

(1, 2)

Hence the part of V in ® must be either the set ° 0 ° or the set — • — • —
indicated in Figure 73. (The line — • — • — • — can be in any position

THREE-PERSON GAME

409

below the points ° °.) However — • — • — • — must be subjected to all
permutations of 1,2,3, — i.e. to rotations of
the triangle by 0°, 60°, 120°, — to produce
all solutions. (Cf. 32.2.3., ° 0 ° is (32 :B),

— • — • — is (32 :A) there.)

Having found the part of V in ®, we
proceed to determine the remainder of V.

Since V is a solution, this remainder must
lie in the area which is undominated by the
part of V in ®. Comparison of Figure 73
with Figure 72 shows that this undominated
area is the following one:

For the set ° 0 ° it consists of the three ^ triangles of Fig. 74, for the
set — — it consists of the three triangles of Figure 75. 1

It is clear from Figure 72, that no point in any one of these triangles
can be dominated by a point in another one. 2 Hence the condition (44:E:c)
of 44.7.3., which characterizes the solution V for E(e 0 ), i.e. for the entire
fundamental triangle — and which holds for the part of V in ® taken for

Figure 74. Figure 75.

® (in place of the entire fundamental triangle, i.e. F(e 0 )) too — states pre-
cisely this: (44:E:c) holds for the part of V in each triangle ik, taken
for that triangle.

47.5. Case (IV) : Second Part

47 . 5 . 1 . Let us therefore take one of those triangles, denoting it by T.
Its position in the fundamental triangle, 8 and the shaded areas dominated

1 The position of all these triangles are clearly indicated by the drawings, except for
the lower triangle in Figure 75. This triangle lies certainly outside the inner triangle
(area Q)— this is equivalent to the restriction (32:8) in 32.2.2., cf. also Figure 60 there.
Its position with respect to the outer (fundamental) triangle is less definite: It may shrink
to a point or even disappear altogether.

It is not difficult to see that the latter phenomenon is excluded, unless the (linear)

2eo 1 / «o\

size of the inner triangle is ^ i of the outer one-— this means I - g 4 ^ + 3

i.e. e 0 & 1. We do not propose to discuss this subject further.

* All this refers to Figure 74, or all to Figure 75 — but, of course, never to both in the
same argument!

* Up to a rotation by 0°, 60°, or 120°. For the lower triangle of Figure 75 the apex
does not lie on the inner triangle, but below it, (cf. footnote 1 above) but this does not
alter our discussion.

410 COMPOSITION AND DECOMPOSITION OF GAMES

by a given point in it (taken over from Fig. 72) are shown on Figure 76.
We may now restrict ourselves to this triangle T, and to the concept of
domination which is valid in it — and determine the solution of (44:E:c)
with respect to this. We redraw T and the setup in it separately, also
introducing a system of coordinates z, y in it. (Figure 77.)

Note that the apex o is undominated by points of T hence it must
belong to V. 1,2

Figure 76.

Figure 77.

Figure 78.

47 . 5 . 2 . Now consider two points of V in T at different heights y. In
order that the upper one should not dominate the lower one, the latter must
not lie in the two shaded sextants, belonging to the former, i.e. the lower
point must be in the middle sextant below the upper one, and vice versa .
Thus, if a point of V in T is given, then all points of V in T at different
heights y must lie in one of the two sextants Uil indicated in Figure 78.

1 For other triangles (i.e. T) than the lower one of Figure 76, this follows from

another consideration, too: As Figures 74, 75 show, the apex of such a triangle lies on the
border of the inner triangle (area ®) and belongs to what we know to be the part of
V in ®.

1 When the lower triangle (i.e. T) of Figure 76 degenerates to one point (cf.
footnote 1 on p. 409) which is, of course, o- — then this determines the part of V in T.

THREE-PERSON GAME

411

47.5.3. Now assume that a y i is the height of more than one point of V.
Let then p and q be two different points of V with this height y x (Figure 79).
Now choose a point r in the interior of the triangle Comparison

of Figure 79 with Figure 77 shows that this r dominates both p and q. As
p, q belong to V, r cannot belong to it. Hence there must exist a point *
in V which dominates r. Now a second comparison of Figure 79 with
Figure 77 shows that a point which dominates r must also dominate either
p or q. Since s, p, q all belong to V this is a contradiction.

Figure 79.

Figure 80.

47.5.4. Next assume that a y x (in triangle T, i.e. between the base l and
the apex o) is the height of no point of V. There exist certainly points of
V with heights y ^ y h e.g. the apex o is such a point. Choose a point p
of V with a height y t/i as low as possible, i.e. with its y minimum. 1
(Figure 80.) Denote this minimum value with y = t/*. Clearly y i < y%.
By the definition of y% no point of V has a height y with y x ^ y < y% and
by the above p is the only point of V with a height y = p*.

Now project p perpendicularly on y = y i, obtaining q. q cannot be in V
hence it is dominated by an $ in V. Hence this s cannot lie below q, i.e.

1 This is possible since V is a closed set. Cf. (*) of footnote 1 on p. 384.

412 COMPOSITION AND DECOMPOSITION OF GAMES

its height y y%. Consequently y y*. Comparison of Figure 80 with
Figure 77 shows that p does not dominate q. Hence necessitating

y 9* y%. Thus y > y% } i.e. 8 lies (definitely) above p. Now a second
comparison of Figure 80 with Figure 77 shows that if a point 8 above p
dominates q , then it must also dominate p. Since s, p both belong to V>
this is a contradiction.

47 . 5 . 5 . Summing up: Every y (between l and 6) is the height of precisely
one point of V. If y varies, then this point changes within the restrictions

Figure 82. Figure 83.

of Fig. 78., i.e. without leaving the sextants HH indicated there. In other
words:

(47 :6) V (in T) is a curve from o to Z, the direction of which never

deviates from the vertical by more than 30 01 (cf. Figure 81).

Conversely, if any curve according to (47:6) is given then comparison
of Figure 81 and Figure 77 makes it clear that the areas dominated by
the points of V sweep out precisely the complement of V in T. So (47 :6)
is the exact determination of the part of V in T*

We can now obtain the general solution V for E(e 0 ) (i.e. for the funda-
mental triangle) by inserting curves according to Figure 81 into each triangle

1 Hence it is continuous.

* It is equally true when T degenerates to a point, cf . footnote 1 on page 409.

THREE-PERSON GAME 413

A of Figures 74 and 75. The results are shown on Figures 82 and 83,
respectively. 1

It will be observed that these figures show still marked similarity with
those pertaining to the solutions of the essential three-person game in the
old theory (cf. 32.2.3., shown in the inner triangle of Figure 73). The new
element consists of the curves in the small triangles, all of which are situated
in the fringe between the two major triangles of Figures 82 and 83. The
width of this fringe, as shown in Figure 71, et sequ., is measured by e 0 . 2
So when e 0 tends to zero, our new solutions tend to the old ones.

It is also worth pointing out that the variety of the solutions is much
greater now than ever before: Entire curves can be chosen freely (within
the limitations of (47:6) above). We will see later, that these curves
motivate an interpretation which is of further significance. (Cf. 47.8.)

47.6. Case (V)

47 . 6 . 1 . Case (V): He 0 <3. In this case 1- jgO<l+|a D d

< 1 + These inequalities express, as is easily verified,

that the orientation of the inner triangle of Figure 71 is inverted, but that it is
still situated entirely within the outer (fundamental) triangle, as indicated
on Figure 84. The latter is again subdivided into seven areas, each of which
can be characterized by stating which two element sets are effective in it
in the sense of (47:5) in 47.2.3. The only difference between the present
situation and that one in Case (IV) (i.e. Figure 71) is the behavior of area ®.
The list is given below Figure 84.

Now we can draw the analogue of Figures 54 and 72, indicating for each
point of the fundamental triangle the shaded areas 4 which it dominates.
This is done in Figure 85, according to (47:5).

It is clear from Figure 85 that no point of the area ® 5 is dominated by
any point. 6 Hence V must contain all of ®.

1 The lower triangle of Figure 83 may degenerate to a point or even disappear alto-
gether, cf. footnote 1 on p. 409.

* The sides of the outer (fundamental) triangle are given by a* « — ^1 + those
of the inner triangle by a' — ^1 — ^ (Cf. Fig. 71). The difference of — ^1 +

and

- 0 - t) * e »-

* This latter inequality is equivalent to e Q
4 Excluding their boundaries.

6 Including its boundary.

<

3.

• I.e. by any a in E(e 0 ). It is easy to show that they are dominated by no a at all
— they are the detached imputations, by (45 :D) in 45.2.4.

The points of the interior of the area ® dominate no other points either. I.e. they
— ► *

dominate no a in E(e 0 ). Again, it is easy to show that they dominate no a at all —
they are the fully detached imputations, cf. (45 :C) in 45.2.4.

These statements can also be verified directly, by using the definitions of 45.2.

THREE-PERSON GAME

415

47.6.2. Having found the part of V in ®, we proceed to determine the
remainder of V. Since V is a solution, this remainder must lie in the area
which is undominated by the already known part of V> i.e. by ®. Con-
sideration of Figure 85 shows that this undominated area consists precisely
of the three triangles ©, ®, 0. 1

It is clear from Figure 85 that no point in any of the three triangles
can be dominated by a point in another one. Hence the argument
of 47.4.2. shows, that our requirement of V must be precisely this:
(44:E:c) of 44.7.3. must hold for the part of V in each one of these triangles,
taken for that triangle (in place of
the entire fundamental triangle, i.e.

E(e 0 )).

The conditions in the triangles ©,

®, © are the same as those described
in Figures 76, 77 for the triangle T.

Hence the entire deduction of 47.5.1.-
47.5.4. may be repeated literally, and
the parts of V in ©, ®, ® are curves
as shown on Figure 81, characterized
by (47:6) in 47.5.5.

V. The triangle Wm
and the curves S

Figure 86.

We can now obtain the general solution V for E(e 0 ) (i.e. for the funda-
mental triangle) by inserting such curves into ©, ©, ® in Figure 85. The
result is shown in Figure 86. For further remarks concerning these solu-
tions cf. 47.8., 47.9.

47.7. Case (VI)

47.7. Co ^ 3.

In this case

2e 0

3

< 0 <

1 + 7 ^ and

-2

These inequalities express, as is easily verified,

that the inner triangle of Figure 84 has still the same orientation, but that
it reaches the boundaries of the outer (fundamental) triangle, and possibly
beyond,* as indicated on Figure 87. The only difference between the present
situation and that one in Case (V) (i.e. Figure 84) is the disappearance of
the areas ©, ®, ®. The list is given below Figure 87.

The analogue of Figures 54. 72 and 85 indicating the domination rela-
tions, is contained in Figure 88.

The argument of 47.6.1. can be repeated literally, proving that V con-
tains all of ®. Consideration of Figure 88 shows that ® leaves no part
of the fundamental triangle undominated. 4 Hence V is precisely ®. For
further remarks concerning this solution, cf. 47.9.

1 The remainder of the fundamental triangle is dominated by the boundary of ®
which belongs to (T).

* This last inequality is equivalent to e 0 3.

8 When e 0 > 3.

4 The remainder of the fundamental triangle is dominated by the boundary of ® which
belongs to ®.

416 COMPOSITION AND DECOMPOSITION OF GAMES

47.8. Interpretation of the Result:

The Curves (One Dimensional Parts) in the Solution

47 . 8 . 1 . The solutions obtained in the discussions of 47.2.-47.7. deserve
a brief interpretative analysis. It is quite conspicuous that the repeated

*

Area:

Effective two-element
sets S :

®

(2, 3)

®

(1, 3)

©

(1, 2)

appearance of a small number of qualitative features goes far in character-
ising their structures — insofar as they deviate from the types familiar in

THREE-PERSON GAME

417

the solutions of the essential three-person game of the old theory. These
features are: The curves — arbitrary within the restriction (47:6) of 47.6.5. —
which occur as soon as e 0 > 0 (and as long as e 0 < 3) ; and the two-dimen-
sional areas, which appear when e 0 > f . We will now undertake their
interpretation.

Consider first Case (IV): 0 < e 0 < f (in the “normal” zone). Let us
consider those solutions of the present case which extend the non-discrimina-
tory solution of the old theory (cf. 33.1.3. and (32 :B) in 32.2.3.). Such a
solution is pictured on Figure 82.

This figure shows the three points ° which form the analogue of a solu-
tion in the old theory. Taking, e.g. the lower point °, one verifies easily
that there

i.e.

OC\ = — 1 + €o, Ot 2 = a 3 = 4-

Thus these three points express an arrangement where two players have
formed a coalition, obtained its total proceeds (amounting to 1), and divided
them evenly — but the defeated player has not been reduced to his minimum
value — 1, because he retained beyond that the total available excess e 0 .

Now the curves, starting from these points ° (in the fringe between the
two triangles), express the situation where the total excess e 0 is not left
in the indisputed possession of the defeated player. By claiming any part
of the excess, the victorious coalition exacts more than the amount 1
which it can actually get in the game — i.e. it ceases to be effective. (Cf.
the areas ©, ®, ® in the Figures 71 and 72.) Therefore the conduct
of affairs of this coalition — the distribution of the spoils within it — is no
longer determined by the realities of the game — i.e. by the threats between
the partners — but by the standard of behavior. This is expressed by the
curve, which is part of the solution. The possible threats between the
partners still restrict this curve to a certain extent (cf. (47:6) in 47.5.6.),
but beyond that it is highly arbitrary. It must be re-emphasized that this
arbitrariness is just an expression of the multiplicity of stable standards of
behavior — but a definite standard of behavior, i.e. solution, means a definite
curve, i.e. rule of conduct in this situation.

47 . 8 . 2 . These considerations suggest the following tentative interpreta-
tion:

(47 :A) In the presence of a positive excess it may happen that a

coalition can obtain beyond its effective maximum also some
fraction of the excess. This possibility is then due entirely
to the standard of behavior and not to the physical possibilities
of the game. The fraction of the excess thus obtained may vary
from 0% to 100% and be left undetermined by the standard of
behavior. The latter will prescribe, however, uniquely, how

418 COMPOSITION AND DECOMPOSITION OF GAMES

the fraction obtained is to be distributed, between the members
of the coalition. This rule of division will depend on which o*
the many possible stable standards of behavior is chosen, and
if the latter is varied, this rule will vary widely, although not
quite unrestrictedly.

We have seen already, that undetermined curves according to (47:6)
occur in many solutions, and they will occur again in the future. The above
interpretation seems to fit them in every case.

The indefiniteness of the distribution of the excess between the victorious
coalition and the defeated player (in a given solution) is an instance how
certain social adjustments may be left open even within a specified social
order. Our curves express the further nuance that while such an indefinite
distribution is decided upon, some players can be tied to each other by
definite conventions. (We will see further instances of this in the third
remark of 67.2.3, 67.3.3. and in 62.6.2.)

47.9. Continuation : The Areas (Two-dimensional Parts) in the Solution

47 . 9 . 1 . The interpretation (47 :A) in 47.8.2. could be tested by applying
it to the extension of the discriminatory solution of the old theory (cf. 33.1.3.
and (32 :A) in 32.2.3.) as pictured on Figure 83. This would bring up some
instructive view-points, particularly with respect to the curve in the lower
triangle of Fig. 83. However, we refrain from elaborating this case any
further.

We turn, instead, to the Cases (V) and (VI), specifically when e 0 > f
(these are the “too large” excesses in the sense of 44.6.1., 45.2). These
cases are characterized by the circumstance, that their solutions contain
two-dimensional areas. Actually, two different situations may arise:

(a) Case (V), i.e. | < e 0 < 3. A solution V contains the two-dimen-
sional area ®, but besides also curves as discussed in 47.8. (cf. Fig. 86).

(b) Case (VI), i.e. e 0 ^ 3. The unique solution V is the two-dimen-
sional area ®, and nothing else (cf. Fig. 88).

The emergence of two-dimensional areas within the solution indicates
that the standard of behavior fails to contain rules of distribution at least
within certain limits. In the Cases (a), (b) these limits are specified. In
the case (a) the curves of 47.8. appear outside of those limits, i.e. the
standard of behavior still sanctions certain coalitions — in the Case (b) this
is no longer the case.

47 . 9 . 2 . So we see that the “disorganizing” effect of a “too large”
excess — i.e. gift from an outside source (cf. 44.6.1.) — manifests itself in two
successive stages: In the Case (a) it is present in a certain central area, but
does not exclude certain conventional coalitions. In the Case (b) the
standard of behavior no longer allows coalitions but it sets certain limiting
principles for the distribution.

THREE-PERSON GAME

419

We have seen that^these successive stages of disorganization are reached
at 6 0 = i f 3 respectively. 1

These considerations seem to be quite instructive in a qualitative way
for the possibilities of standards of behavior and organizations. It appears
likely that they will provide useful guidance in the further development
of the theory. But the reader must be cautioned against drawing far
reaching conclusions from the quantitative results: They apply to the
three-person game with an excess, 2 which is thus shown to be the simplest
model for their operation. But it must have become amply clear by
now that an increase in the number of participants will affect conditions
fundamentally.

1 Note that |r|i — f .

2 Hence also to a decomposition six-person game in the old theory, cf. 46.12.

CHAPTER X
SIMPLE GAMES

48. Winning and Losing Coalitions and Games Where They Occur
48.1. The Second Type of 41.1. Decision by Coalitions

48.1.1. The program formulated in 34.1. provided for far-reaching
generalizations of the games corresponding to the 8 corners of the cube Q,
introduced in 34.2.2. The corner VIII (also representative of II , ///, IV)
was taken up in 35.2.1. and provided the source for a generalization, lead-
ing to the theory of composition and decomposition to which all of Chapter
IX was devoted. We now pass to the corner I (also representative of V ,
VI, VII), which we will treat in a similar fashion.

By generalizing the principle, of which a special instance can be dis-
cerned in this game, we will arrive at an extensive class of games, to be
called simple. It will be seen that a study of this class yields a body of
information which is of value for a deeper understanding of the general
theory in the sense of 34.1.

48.1.2. Consider the corner I of Q, discussed in 35.1. As was brought
out in 35.1.1., this game has the following conspicuous feature: The aim of the
players is to form certain coalitions consisting either of player 4 and one ally,
or of all three other players together. Any one of these coalitions is winning
in the full sense of the word. Any coalition which falls short of these is
completely defeated. I.e. the quantitative element, the payments expressed
by the characteristic function, can be treated as something secondary — the
primary aim in this game is to succeed in forming certain decisive coalitions.

This description suggests strongly that the number four of players
and the particular scheme of decisive coalitions are special and accidental
and that a more general principle can be extracted from this particular
arrangement.

48.1.3. In carrying out this generalization, the following observation is
useful. In our above example, the decisive coalitions — the attainment of
which is the sole aim of the players — were these:

(48:1) (1,4), (2,4), (3,4), (1,2,3).

Now it is convenient to view not only these as winning coalitions, but also
all their (proper) supersets:

(48:2) (1,2,4), (1,3,4), (2,3,4), (1,2, 3, 4).

The point is that although the coalitions (48:2) contain participants whose
presence is not necessary in order to win, the coalition is nevertheless a

420

WINNING AND LOSING COALITIONS

421

winning one — i.e. the opponents are defeated. 1 These opponents form
those coalitions which are the complements of the sets in (48:1), (48:2), i.e.
the sets

(48:3)

(2,3), (1,3), (1,2), (4).
(3), (2), (1), ©.

Thus (48:1), (48:2), contain the winning coalitions, and (48:3) contains
the defeated ones.

It is easily verified that every subset of 7 = (1,2, 3, 4) belongs to pre-
cisely one of these two classes: (48:1), (48:2), or (48:3). 2

48.2. Winning and Losing Coalitions

48 . 2 . 1 . Let us now consider a set of n players: I = (1, • • • , n). The
scheme of 48.1.3. generalizes to subdividing the system of all subsets of I
into two classes W and L, such that the subsets of W will represent the
winning coalitions and the subsets of L will represent the losing ones. The
analogues of the properties formed in 48.1.3. can be formulated as follows:

Denote the system of all subsets of 7 by 7. 3 The mapping of every
subset S of / on its complement (in I ) :

(48:4) S-+-S

is clearly a one-to-one mapping of 7 on itself. Now we have:

(48 :A :a) Every coalition is either winning or defeated and not both —

i.e. W and L are complementary sets in 7.

(48:A:b) Complementation (in I) carries a winning coalition into a

losing one and vice versa — i.e. the mapping (48:4) maps W and
L on each other.

(48:A:c) A coalition is winning, if part of it is winning — i.e. W con-

tains all supersets of its elements.

(48:A:d) A coalition is losing, if it is part of one which is losing — i.e.

L contains all subsets of its elements.

48 . 2 . 2 . Before we discuss the concepts of winning and losing in their
relationship to the game, we may analyze the structure of conditions
(48:A:a)-(48:A:d) somewhat further.

The first conspicuous fact is that, although we need both classes W and L
to interpret the game, these classes determine each other. Indeed they do
this in two ways: Given one of W or L (48:A:a) as well as (48:A:b) can be
used to construct the other. I.e. starting from one of these sets, the other
one is obtained in this way:

According to (48 : A :a) : Take the given set as a whole and form its comple-
ment (in 7).

1 I.e. the complements are flat in the sense of 31.1.4. Cf. the discussion in 35.1.1.

* (1,2, 3, 4) has 2 * — 16 subsets, of these 8 are in (48:1), (48:2), and the remaining 8 in
(48:3).

* As / has n elements, 7 has 2" elements.

422

SIMPLE GAMES

According to (48:A:b): Take each element of the given set separately,
and replace it by its complement (in J). 1

It should be noted also that if the given set, W or L, possesses the prop-
erty (48:A:c) or (48:A:d) respectively, then the other set — obtained from
the former by (48:A:a) or by (48:A:b) will possess the other property
(48:A:c) or (48:A:d). 2

It follows from the above, that we can base the entire structure now
under consideration on either one of the two sets W and L. We must only
require that both transformation (48:A:a) and (48:A:b) lead from it to the
same set (which is then the other one of W and L) and that it must satisfy
the pertinent one of the two conditions (48: Arc) and (48:A:d) (the other
condition of (48: Arc) and (48:A:d) is then automatically taken care of,
according to what we have seen).

Thus we have only two conditions for W or L : First the equivalence of
(48:A:a) and (48:A:b) and second (48: Arc) or (48:A:d).

The former condition means this: The non-elements of the set coincide
with the complements (in I) of the elements of the set. In other words: Of
two complements (in I) S , — S, one and only one belongs to the set.
Summing up :

The sets W(z I) are characterized by these properties:

(48 :W)

(48:W:a) Of two complements (in I) S, —S, one and only one

belongs to W.

(48:W:b) W contains all supersets of its elements.

The sets L(£ I) are characterized by these properties:

(48 :L)

(48:L:a) Of two complements (in I) S, —S, one and only one

belongs to L.

(48:L:b) L contains all subsets of its elements.

1 The reader will note the remarkable structure of this condition : The given set must
produce the same result, irrespectively of whether complementation is applied to it as a
unit, or to its elements separately.

1 This is actually true for (48:A:a) as well as for (48:A:b), and independent of the
question whether (48:A:a) and (48:A:b) produce the same set. Precisely:

(48 :B) Let a set M possess the property (48:A:c) [ (48: A:d)), then both sets which

are obtained from it by (48:A:a) and (48:A:b) — we do not assume that they
are identical — possess the other property (48:A:d) [(48:A:c)].

Proof : We must show that both transformations (48:A:a) and (48:A:b) carry (48:A:c)
into (48:A:d) and vice versa.

Clearly (48:A:c) is equivalent to this:

(48:A:c*) If S is in Af, and T is not, then S £ T is excluded.

Again (48:A:d) is equivalent to this:

(48:A:d*) If S is not in Af, and T is, then S £ T is excluded.

Now the transformation (48:A:a) interchanges “being in M ” and “not being in M ” with
each other. Hence it interchanges (48:A:c*) and (48:A:d*). The transformation
(48:A:b) interchanges £ and 2 (this is brought about by individual complementation for
the elements S, T; besides the symbols S, T must be interchanged). Hence it, too,
interchanges (48:A:c*) and (48:A:d*).

CHARACTERIZATION OF THE SIMPLE GAMES

423

We restate:

If W [L] fulfills (48:W) [(48:L)], then (48:A:a) and (48:A:b) yield the
same setL [W]. W&ndL fulfill (48:A:a)-(48:A:d) and L [W\ fulfills (48:L)
[(48:W)]. Conversely, if W, L fulfill (48:A:a)-(48:A:d), then they fulfill
separately (48 :W), (48 :L).

49. Characterization of the Simple Games
49.1. General Concepts of Winning and Losing Coalitions

49.1.1. We now pass to the consideration of the connection between
winning and losing coalitions in the game itself.

Assume, therefore, that an n-person game r is given. In all the con-
siderations which follow, it is advantageous to restrict ourselves to the old
theory in the sense of 30.1.1. or 42.4.1. Consequently, as pointed out in
42.5.3., we may assume T to be zero or constant-sum as we desire. For the
present we prefer to choose r as a zero-sum game.

Beyond this r is not restricted and in particular no normalization is
assumed.

49.1.2. Let us first analyze the concept of a losing coalition. Repeating
essentially what was said in 35.1.1., we may argue as follows: 1 The player i,
when left to himself, obtains the amount v((i)). This is manifestly the
worst thing that can ever happen to him, since he can protect himself
against further losses without anyone else’s help. Thus we may consider
the player i when he gets this amount v((i)) to be completely defeated. A
coalition S may be considered as defeated, if it gets the amount ^ v((i)),

»' in S

since then each player i in it must necessarily get v((i)). 2 Thus the criterion
of defeat is

v(S) = X v((i)).

i in S

In the terminology of 31.1.4., this means that the coalition S is flat. (Cf.
also footnote 3 on page 296.)

We have obtained a satisfactory definition of the system Lr* of all losing
(defeated) coalitions:

(49 :L) Lt is the set of all flat sets S(s I).

It is now easy to say what a winning coalition is. It is plausibly one,
the opponents of which are losing, i.e. the system Wr* of all winning coali-
tions is this:

1 The difference is that our present r is more general.

1 Since no player i need ever accept less than v((t)), and those in the coalition 8 have

together ^ v ((»)), this is the only way in which they can split.
iinS

* In order to avoid confusion, we will use the symbols W r, Lr instead of the W y L of
48.2.2. The difference between this and the former is that 48.2.2. is a postulational
discussion of the properties which appeared desirable for the concepts of “winning”
and " losing ’ * (described by W , L) — while we are now analyzing definite sets obtained
from a specific game r.

The two viewpoints will be merged in (49 :E) of 49.3.3.

424

SIMPLE GAMES

(49 :W) Wr is the set of all sets S(z I) for which — S is flat.

It should be conceptually clear, and is also immediately verified with
the help of 27. 1.1. -2., that the sets Wr, Lr are invariant under strategic
equivalence.

49.1.3. We cannot expect the above Wr, Lr to fulfill the conditions
(48:A:a)-(48:A:d) (for W, L) of 48.2.1. The game in its present generality
need not be of the simple type referred to, where the only aim of all players
is to form certain decisive coalitions and there are no other motives which
require a quantitative description. 1 It will therefore be necessary to
restrict in order to express the property we have in mind. Indeed, the
precise formulation of this restriction is our immediate objective.

Nevertheless, we begin by determining how much of (48:A:a)-(48:A:d)
holds true for the T in its present generality. We give the answer in several
stages.

(49 :A) Wr, L r always fulfill (48 : A :b)-(48 : A :d)

Proof: Ad (48:A:b): Immediate by comparing (49 :L) and (49 :W) in
49.1. 2. 2

Ad (48:A:c), (48:A:d): Since we have (48:A:b), we can apply (48:B) in
48.2.2. 8 and therefore (48:A:c) and (48:A:d) imply each other..

But (48:A:d) coincides with (31:D:c) in 31.1.4., considering (49:L).

Thus the main difference between our present W r , Lr and the setup of
48.2. lies in (48:A:a) — i.e. in the question whether or not Wr and Lr are
complements. We can decompose this assertion into two parts:

(49:1)

(49:1 :a) W r nL r = ©, 4

(49:1 :b) W r uL r = 7. 5

(49:l:a) leads back to familiar concepts:

(49 :B)

(49:B:a) (49:1 :a) holds if and only if T is essential.

(49:B:b) If T is inessential, then Wr = Lr = L 6

1 Our discussion of the four-person game has provided many illustrations of such
motives, for which the end of 36.1.2. provides a good instance. This situation is, indeed,
the usual (general) one — the class of games at which we are aiming now, is in a certain
sense an extreme case, cf. the concluding observation of 49.3.3.

* Actually the concept of “winning” was based on the concept of “losing” by just
this operation of complementation.

*It appears now why we separated (48:A:a) from (48:A:b) in 48.2.1: We have now
(48:A:b), but not (48:A:a).

4 It may seem odd that this — no coalition can at the same time be both winning and
losing — must be stated separately. The meaning of this condition will appear in (49 :B)
and footnote 6.

‘This states that every coalition — i.e. every subset of I — is definitely winning or
losing. This is, of course, the idea on the basis of which we wish to specialize r.

• Thus a coalition can at the same time be both winning and losing, when the game is
inessential — manifestly because in this case both states are irrelevant.

CHARACTERIZATION OF THE SIMPLE GAMES

425

Proof: Ad (49:B:a): The negation of (49:1 :a) is the existence of an S
such that both S and — S are flat. This amounts to inessentiality by
(31 :E :b) in 31.1.4.

Ad (49 :B :b) : Wr = Lr = I means that every S in I is flat. This amounts
to inessentiality by (31:E:c) in 31.1.4.

Before passing to (49:1 :b) we note that Wr, Lr possess one property
which did not occur in (48:A:a)-(48:A:d).

(49 :C) Lr contains the empty set and all one-element sets. 1

Proof : This coincides with (31:D:a), (31:D:b) in 31.1.4.

(49 :C) is really a new condition, i.e. it is not a consequence of (48:A:a)-
(48:A:d); we will verify this in 49.2. below. Thus our plausible discussion
of 48.2. overlooked a necessary feature of the Wr, Lr . We must, therefore,
make sure that the present conditions contain everything. I.e. that the
conditions (48:A:b)-(48:A:d) and (49 :C), together with the results of (49 :B)
on inessentiality, characterize the W r , L T completely. This will be shown
in 49.3. below.

49.2. The Special Role of One-element Sets

49 . 2 . 1 . We begin with the example announced above: Two systems
W , L which fulfill (48:A:a)-(48:A:d), 2 but not (49 :C). Actually, we can
determine all such pairs.

(49:D) W , L fulfill (48:A:a)-(48:A:d), but not (49:C), if and only if

they have the following form : W is the set of all S containing i 0f L
is the set of all S not containing i 0 , where i 0 is an arbitrary but
definite player.

Proof: Sufficiency: It is immediately verified that the W, L formed as
indicated fulfill (48:A:a)-(48:A:d). (49 :C) is violated, since the one ele-

ment set ( i 0 ) belongs to W, and not to L.

Necessity: Assume that W, L fulfill (48:A:a)-(48:A:d), but not (49:C).
Let ( i 0 ) be a one-element set, which does not belong to L. 3 Then (t 0 )
belongs to W .

Every S containing i 0 has S 2. (to), hence it belongs to W by (48:A:c).
If S does not contain t 0 , then — S contains it; hence — S belongs to W by
the above and S belongs to L by (48:A:b).

Finally W, L are disjunct by (48:A:a), hence W is precisely the set of the
S containing i 0 and L is precisely the set of the S not containing t 0 .

49 . 2 . 2 . It may be worth while to comment briefly upon this result.

1 It is clearly in the spirit of our entire analysis of games that a coalition of one player
is to be considered as defeated — as this player has not succeeded in finding partners for a
coalition.

2 We mentioned originally (48:A:b) — (48:A:d) only, but the above strengthening
requires no extra effort.

* If the empty set does not belong to L, then no set can belong to L owing to (48:A:d) ;
hence any (to) will do.

426

SIMPLE GAMES

The W , L formed in (49 :D) cannot be the W r, Lr of any game since
they violate (49 :C). This may seem odd, since (49 :D) appears to convey a
very clear idea of the kind of “ winning ” and “ losing” described by its
W, L. Indeed, they describe the situation where a coalition wins if the
player i 0 belongs to it, and loses if he does not. Why can no game be con-
structed to fit this specification?

The reason is that under the conditions described, “winning” would
not be a matter of forming coalitions at all : 1 The player i 0 is “victorious”
without anybody else’s help. Still worse, in our terminology this position
of to is no victory — it is not the result of any strategic operation , 2 but a
fixed state given him by the rules of the game . 8 A game in which coalitions
involve no advantage is inessential, 4 — even if one player t 0 should have a
considerable fixed advantage in it.

The reader will understand, of course, that all this is just an additional
comment on results which were already rigorously established above (in
(49 :C), (49:D)).

49.3. Characterization of the Systems W f L of Actual Games

49 . 3 . 1 . We now turn to the second subject mentioned at the end of
49.1.3. Let two systems W y L(zl) be given, which fulfill the conditions
(48:A:b)-(48:A:d) and (49:C), and also (49:1 :a ). 6 We wish to construct an
essential game T with Wr = W, Lr = L. In doing this, we normalize T
with 7 = 1 .

The sets S in Lr are characterized by their flatness, i.e. by v(S) = — p,
where p is the number of elements of £. 8 The sets S in W r are character-
ized by the fact that —S belongs to L r , i.e. by v( — S) = — (n — p), owing
to the above. Now v(-S) = — v(S), hence we may write for this
v(S) = n — p.

Hence we have shown:

The desired relations W T = W, L r = L are equivalent to this:

(49:2) For a g-element set S, (q = 0, 1, • • • , n — 1, n)

(49:2:a) v(S) = n — q

if and only if S belongs to W, and
(49:2:b) v(S) = -q

if and only if S belongs to L.

1 The equivalent consideration was carried out in a special case in 35.1.4.

1 We always consider this to be the same thing as forming appropriate coalitions.

a Cf. our treatment of the basic values a', 6', c' in the three-person game, in 22.3.4.
The entire discussion of strategic equivalence, cf. 27.1.1., was made in the same spirit:
advantages like this one can be removed by strategically equivalent transformations,
while those which are really due to forming coalitions, cannot.

4 Hence its Wr, Lv are not the desired ones, described in (49 :D), but those of (49:B:b).

* We require (49:1 :a) because we aim primarily at essential games (cf. (49 :B)). Sub-
sequently we will make our discussion exhaustive — as will be seen in (49 :E).

• Recall that all v ((f)) ■» —7 — — 1 .

CHARACTERIZATION OF THE SIMPLE GAMES 427

Thus our task is to construct a game r (normalized and 7 = 1) with a
characteristic function v(S) which fulfills (49:2).

49 . 3 . 2 . (49:2) determines v(S) for the S of W and L, so we need only
define it for those S which belong to neither set. We try there the value 0.
Accordingly we define :

S a ^-element set with q = 0, 1, • • • , n — l,n.

\ 0 otherwise 1

We first prove that v(S) is a characteristic function, i.e. that it fulfills
(25:3:a)-(25:3:c) in 25.3.1. We prove these conditions in their equivalent
form of (25 :A) in 25.4.2.:

Case p = 1 with = : This is v( I) =0, immediate since 7 is in W, because
© = —7 is in L by (48:A:b), (49:C).

Case p = 2 with = : This is v(Si) + v(S 2 ) = 0 when Si, S 2 are comple-
ments. If both Si, S 2 are not in W, L, then v(Si) = v(S 2 ) = 0. If one of
Si, S 2 is in W or L, then the other is in L or W, respectively, by (48:A:b).
Assume, by symmetry, Si in L, S 2 in W. Let Si have q elements, hence
S 2 has n — q. Then v(Si) = — q } v(S 2 ) = q.

So at any rate v(Si) + v(S 2 ) = 0.

Case p = 3 with ^ : This is v(Si) + v(S 2 ) + v(S 3 ) ^ 0 when Si, S 2 , Sj
are pairwise disjunct with the sum 7. If none of Si, S 2 , S 8 is in W, then
v(£i), v(S 2 ), v(S 3 ) ^ 0. 2 If one of Si, S 2 , S 3 is in W, we may assume by
symmetry , that it is S 3 . Hence — S 8 = Si u S 2 isinLby (48:A:b), andsoSi,

5 2 are in L by (48:A:d). Let Si have q x elements, S 2 have q 2 elements, hence

5 3 has n - q x - q 2 . Then v(Si) = -q h v(S 2 ) = - q 2) v(S 3 ) = q x + q 2 .

So at any rate v(Si) + v(S 2 ) + v(S 3 ) ^ 0.

49 . 3 . 3 . Thus v(S) belongs to a game T. We now establish the remaining
assertions.

v(S) (i.e. T) is normalized and y = 1 : Indeed, by (49 :C) all v((i)) = — 1.
v(S) fulfills (49:2): Owing to (48:A:b) and v( — S) = — v(S), the two
parts of (49:2) go over into each other if we interchange S and — S. We
consider therefore only the second half.

If S is in L, then clearly v(S) = —q. If S is not in L, then v(S) = — q
would necessitate 0 = — q? or q = 0. But this means that S is empty,
contradicting (49 :C).

So the game V possesses all desired properties.

We are now able to prove the following exhaustive statement:

(49 :E) In order that two given systems IF, L(s7) be the Wr, Lr

of a suitable game T, these requirements are necessary and
sufficient:

T inessential: W = L = 7.

T essential: (48:A:b)-(48:A:d), (49 :C), (49:1 :a).

1 That the two first specifications do not conflict, is due to (49:1 :a).

2 Clearly v(S) ^ 0 if S is not in W.

1 As n — q 7 * — q, S could not be in W; hence v(S) « 0.

j n — q for S in W
v(S) = \ — q for S in L

428

SIMPLE GAMES

Proof: T inessential: Immediate by (49:B:b).

T essential: The necessity was established in (49 :A), (49:B:a), (49 :C).
The sufficiency is the content of the construction which we have carried out.

In concluding we mention another interpretation of (49:2). Recalling
the inequalities (27:7) of 27.2. (also shown on Figure 50), which specify
limitations for v(S), it appears that Wr is the set of those S for which v(S)
reaches the upper limiting value, and L r the set of those S for which v(S)
reaches the lower limiting value.

49.4. Exact Definition of Simplicity

49 . 4 . (49 :E) permits us to give a rigorous definition of that class of
games to which we alluded in 48.1.2. and 48.2.1., and which was circum-
scribed in more detail at the beginning of 49.1.3.: Where the only aim of
all players is to form certain decisive coalitions and where there are no other
involved motives which require a quantitative description.

By combining the part of (49 :E) which refers to essential games with
(49:1), it appears that the formal expression of this idea is

(49:1 :b) W r uL r = L

Indeed, this condition expresses that any given coalition S belongs either
to the winning or to the losing category — without any further qualification.

We define accordingly: An essential game which fulfills (49:1 :b) is
called simple.

The concept of simplicity is invariant under strategic equivalence,
since the sets W r , L r are.

49.6. Some Elementary Properties of Simplicity

49 . 5 . 1 . Before we take up the detailed mathematical discussion of this
concept, let us consider once more the closing remark of 49.3. In the sense
of that remark an essential game is simple, if v(£) lies for every S on the
boundary 1 of the area assigned to it by the inequalities (27 :7) in 27.2.

The variety of all essential n- person games (normalized, 7 = 1) can
be viewed as a geometrical configuration of a certain number of dimensions,
given in Figure 65. More precisely the inequalities referred to define a
convex polyhedric domain Q n in the linear space of the dimensionality in
question, and the points of this domain represent all these games. 2

49 . 5 . 2 . E.g.: For n = 3 the dimensionality is zero, and the domain Q*
a single point. For n = 4 the dimensionality is 3, and the domain Q 4
the cube Q of 34.2.2.

Now the simple games are those for which we are on the boundary
of each defining inequality. With respect to the convex polyhedric domain
Q n this means: The simple games are the vertices of Q n) n = 3, 4.

1 The boundary consists of two points: the upper limiting value n — p and the lower
one — p, (7 — 1). v (5) must be one of these two, no matter which.

* The reader who is familiar with n-dimensional linear geometry, will note: Since Q n
is defined by linear inequalities, it is a Dolvhedron. The discussion of 27.6. allows to
conclude that it is convex.

CHARACTERIZATION OF THE SIMPLE GAMES 429

E.g. : For n = 3 Q 3 is a single point, i.e. nothing but a vertex, so the
essential 3-person game is simple. 1 For n = 4 Q 4 is the cube Q , so the simple
games are the vertices, i.e. corners I-VIII . 2

49.6. Simple Games and Their W } L. The Minimal Winning Coalitions : W m

49 . 6 . 1 . Combining (49 :E) with the definition of simplicity, we obtain:

(49 :F) In order that two given systems W> L(s I) be the Wr, Lr

of a suitable simple game T, these requirements are necessary
and sufficient: (48:A:a)-(48:A:d), (49:C).

That the S referred to in (49:2) exhaust all subsets of /, is definitory
for simplicity. Consequently it is for simple games and for these alone,
that knowledge of Wr, L r determines v(S), provided that the game is
normalized and 7 = 1 . I.e., without the last proviso, that it determines

the game up to a strategic equivalence.

We restate this:

(49 :G) In case of simplicity, and only then, the game T is deter-

mined by its W r, L r up to a strategic equivalence.

Consequently, according to (49 :F) and (49 :G) a theory of simple games
is coexistensional with the theory of those pairs of systems W, L which
fulfill (48 : A :a)-(48 :A :d) , (49:C).

49 . 6 . 2 . In studying the pairs W } L described above, 48.2.2. should be
recalled, and particularly (48 :W), (48 :L) there and (49:2). According
to these, it is sufficient to name either W or L in order to determine the
pair W y L.

The conditions (48:A:a)-(48:A:d) are then to be replaced as follows:
If W is used, by (48:W); if L is used, by (48 :L).

As to (49 :C), it refers to L directly. We can equally well refer it to W f
by applying (48:A:b) — then the sets mentioned in it must be replaced
by their complements.

For the sake of completeness, we restate (48 :W) and (48 :L), together
with the corresponding forms of (49 :C).

The sets W(s I) are characterized by these properties:

(49 :W*)

(49:W*:a) Of two complements (in I) S f —S f one and only one

belongs to W.

(49 : W * :b) W contains the supersets of its elements.

(49:W*:c) W contains I and all (n — l)-element sets.

1 Cf. also (50: A) in 60.1.1.

1 As far as the comers /, V , VI, VII are concerned, this is no surprise: Our dis-
cussion started with these in 48.1 and our concept of simplicity obtained from them by
generalization.

The reappearance of the corners //, ///, IV y VIII is more puzzling: We treated them
in 35.2. as the prototypes of decomposability. However, they are simple too, as follows
easily from (50:A) and the beginning of 51.6.

430

SIMPLE GAMES

The sets L (£ 1 ) are characterized by these properties:

(49 :L*)

(49:L*:a) Of two complements (in I) S , — S, one and only one

belongs to L.

(49:L*:b) L contains the subsets of its elements.

(49:L*:c) L contains the empty set and all one-element sets.

As pointed out above, we could base the theory on either W with (49 :W*)
or on L with (49 :L*).

49 . 6 . 3 . Since it is more in keeping with the usual way of thinking about
these matters to specify the winning rather than the losing coalitions, we
shall use the first mentioned procedure.

In this connection we observe that a certain subset of W shares the
importance of W. This is the set of those elements S of W of which no
proper subset belongs to W. We call these S the minimal elements of W
(i.e. W r) and their set W m (i.e. W ?).

The intuitive meaning of this concept is clear: These minimal winning
coalitions are the really decisive ones, those winning coalitions in which no
participant can be spared. (It will be remembered that our discussion of
48.1.3. began with the enumeration of these coalitions for the game we were
then considering.)

49.7. The Solutions of Simple Games

49 . 7 . 1 . The heuristic considerations which led us to the concept of
simple games make it plausible, that the discussion of games belonging to
this category, may turn out easier than that of (zero-sum) n- person games
in general. For a corroboration of this we must examine how solutions are
determined in a simple game. Since we are now considering the old form
of the theory, 30.1.1. must be consulted. 1 We begin with the observation
that a considerable simplification must be expected from the fact that in a
simple game every set is either certainly necessary or certainly unnecessary
(cf. 31.1.2.).

49 . 7 . 2 . In order to establish this assertion we prove first:

(49 :H) In any essential game T all sets S of Wr are certainly neces-

sary, and all sets S of Lr are certainly unnecessary.

Proof : If S is in L r then it is flat, hence certainly unnecessary by (31 :F)
in 31.1.5. If S is in W r, then — S is flat (because it is in Lr) and S 7 * Q
(because © is in Lr, hence not in Wr)- So S is certainly necessary by
(31 :G) in 31.1.5.

We can now fulfill our above promise concerning simple games — indeed,
this can be done in two different ways.

1 In the terminology of the new form of the theory — as introduced in 44.7.2. et sequ. —
this means: We are looking for solutions for E( 0), i.e. the excess is being restricted to the
value 0.

The significance. of this restriction will become clearer in the third remark of 51.6.

MAJORITY GAMES AND THE MAIN SOLUTION 431

(49:1) In any simple game r all sets S of W r are certainly necessary,

and all others are certainly unnecessary.

Proof : Combine (49 :H) with the fact, that for a simple game Lr is
precisely the complement of W r.

(49 :J) In any simple game r all sets S of Wp are certainly necessary,

and all others are certainly unnecessary. 1

Proof : We can replace the Wr of (49:1) by its subset W i.e. we can
transfer all S of Wr — Wy from the certainly necessary class into the
certainly unnecessary, owing to (31 :C) in 31.1.3. Indeed, every S in Wr
possesses a subset T in Wp.

Of these two criteria (49:1) and (49 :J), the latter is more useful.
Their importance will be established by actual determination of solutions
in simple games. 2 Indeed, this analysis of simple games permits the
deepest penetration yet effected into the theory of games with many
participants. 3

60. The Majority Games and the Main Solution
60.1. Examples of Simple Games : The Majority Games

60.1.1. Before going any further, it is appropriate to give some examples
of simple games, i.e. of the pairs W, L of (49 :F) in 49.6.1. We know from
49.6.2. that it suffices to discuss the W as characterized by (49 :W*) there.

Let us therefore consider some possible ways of introducing such W — i.e.
possible definitions of a concept of winning.

The principle of majority suggests itself as a particularly suitable
definition of winning. Hence it is plausible to define W as the system of all
those S which contain a majority of all players. It will be noticed, however,
that we must exclude ties — indeed (49:W*:a) states for this W, that for
every S either S or —S must contain the majority of all players, thus
excluding that both may contain exactly half. In other words: The total
number of participants must be odd.

71

So if n is odd, we may define W as the set of all S with > ^ elements. 4

The simple game which obtains in this way, 5 will be called the direct majority
game .

1 Comparison of (49:1) and (49 :J) shows that the S of W r — W? are simultaneously
certainly necessary and certainly unnecessary. This is another illustration for the
remark at the end of footnote 1 on p. 274.

*Cf. 50.6.2. and 55.2.

1 Cf. 55.2.-55.11., and in particular the general remarks of 54.

4 Since the smallest integer > ^ is (n odd!), we may also say: S must have

- elements.

1 Precisely: The class of strategically equivalent ones (of n participants).

432

SIMPLE GAMES

The smallest n for which this can be done, 1 is 3. We know that there
exists only one essential three-person game, and that for this W it consists
precisely of the 2 and 3-element sets — i.e. of the sets with > $ elements.
So we see:

(50 :A) The (unique) essential three-person game is simple; it is

the direct majority game of three participants.

For the subsequent n which are eligible, n = 5, 7, • • • , the direct
majority game is merely one possibility among many.

50 . 1 . 2 . The direct majority game is only available, when n is odd, and
yet simple games exist for even n as well — indeed our prototype of simple
games (cf. 48.1.2., 48.1.3.) had n = 4.

However, the concept of majority is easily extended to cover the case of
even n as well. To this end we introduce weighted majorities in the following
manner: Let each one of the players 1, • • • , n be given a numerical weight ,
say Wi, • • • , w n respectively. Define W as the set of all those S which
contain a majority of total weight. This means:

n

(60:1) X w ' > i X Wi <

% in 8 i — 1

or equivalently,

(50:2) X Wi > X w < ■

% in S i in — S

We must again take care to exclude ties. However, owing to the greater
generality of our present setup, it is better to proceed immediately to a
complete discussion of (49 :W*).

50 . 1 . 3 . Let us see, therefore, what restriction (49 :W*) imposes upon
the wh, • • • , w n .

Ad (49:W*:a): Since we can express that S belongs to W by (50:2), so
— S belongs to W when

(50:3) X < X »<•

t in 5 * in — S

So (49:W*:a) means that always (50:2) or (50:3) holds, but never both.
This means clearly, that never

(50:4) X ««- X Wi >

i in 8 tin —8

or equivalently, that never

(50:5) X w * = * X Wi -

i in 3 i - 1

1 I.e. which is odd and for which a game can be essential.

MAJORITY GAMES AND THE MAIN SOLUTION 433

Ad (49:W*:b): Using the definition of W in the form (50:1), this require-
ment is clearly satisfied if all w { £ 0. 1

Ad (49:W*:c): Using again (50:1), it is clear that I = (1, • • • , n)
belongs to W. For the general (n — l)-element set S = 7 — (to), the
condition (50:1) states that

n

Wi,<i 'Z w < ■

»- 1

Summing up:

(50:B) The weights w h • • • , w n can be used to define by (50:1)

or (50:2) a W which satisfies (49 :W*) if and only if they fulfill
the following conditions:

(50:B:a) For all i 0 = 1, • • • , n

n

0 £ Wi t <i Z/ w '-
<-i

(50:B:b) For all S S I

Z z w <-

i in 8 % - 1

Verbally: A player has always non-negative weight, but never half of
the total weight or more; no combination of players has precisely half of the
total weight. 2

The simple game which obtains from this W z will be called the weighted
majority game (of n participants with the weights ti>i, • • • , w n ). We will
also designate this game by the symbol [w h • • • , w n ].

Thus the direct majority game has the symbol [1, • • • , 1],

It will be noted that the four-person game represented by the corner
I of Q, discussed in 48.1.2., 48.1.3. can be described as a weighted majority
game. Indeed, the principle of winning found in 48.1.3. can be expressed
by saying that players 1,2,3 have a common weight, while player 4 has the
double weight. I.e. this game has the symbol [1,1, 1,2].

60.2. Homogeneity

50 . 2 . 1 . The introduction of majority games and their explanatory
symbols [u>i, • • • , w n ] is a step in the direction of a quantitative (numeri-
cal) classification and characterization of simple games. There are good
reasons to think that it would be most desirable to carry out such a program
fully: Simplicity was defined in combinatorial, set-theoretical terms and
it is to be expected that a numerical characterization would make them

1 This is, of course, a perfectly plausible condition; indeed, the surprising thing is that
we are not forced to require w% > 0 — i.e. that we can permit a weight to vanish.

* The first requirement obviates the difficulty of 49.2., the second excludes ties.

1 Precisely: The class of strategically equivalent ones.

434

SIMPLE GAMES

easier to handle. Such a characterization usually facilitates a more exhaus-
tive, quantitative understanding of the notion considered. Besides, in our
present problem we are ultimately searching for solutions that are defined
numerically, and therefore it seems likely that a numerical characterization
will correspond to them more directly than a combinatorial one.

However, this first step is far from carrying out the transition.

On the one hand, a simple game may possess more than one symbol
[wi y • • • , to*] — indeed, every simple game that has one at all has infinitely
many. 1 On the other hand, we do not know whether all simple games
possess such a symbol at all. 2

We begin by considering the first deficiency. Since the same simple
game may possess several symbols [w i, • • • , w n ]> the natural procedure
is to single out from among them a particular one by some convenient
principle of selection. It is desirable to specify in this principle such require-
ments which increase the significance and usefulness of the w 1 , • • * , w n .

First some preliminary observations. The conditions (50:1), (50:2) sug-
gest consideration of the difference

n

(50:6) a® = 2 2) to,- — £ w 4 = w, — X w< •

i in S i — 1 * in S i in — 8

This a s expresses how much the coalition S outweighs its opponents —
how much of a weighted majority it possesses. These are its immediate
properties:

(50 :C) a s = —as

Proof: Use the last form of (50:6) for a 5 .

(50:D:a) a s > 0 if and only if S belongs to W.

(50:D:b) a s < 0 if and only if S belongs to L.

(50:D:c) a s = 0 is impossible.

Proof : Ad (50:D:a): Definitory.

Ad (50:D:b): Immediate by (50:D:a) and (50 :C).

Ad (50:D:c): Immediate by (50:D:a), (50:D:b) since W, L exhaust all
S. It also coincides with (50:B:b).

50.2.2. Now it is natural to try to arrange the weights Wi, • • • , w n so
that the amount of a s which secures victory be the same for each winning
coalition. It would be unreasonable, however, to require this actually for
all S of W: If S belongs to W, then its proper supersets T do too, and they may
have a T > a s . 8 Since such a T contains participants who are not necessary
for winning, it seems natural to disregard it. I.e. we require the constancy
of as only for those S of W which are not proper supersets of other elements

1 Obviously, sufficiently small changes of the w% will not disturb the validity of (50:1),
particularly since (50:5) is excluded by (50:B:b).

* We will see in 53.2. that certain simple games have none.

* Thus for T — / a S, ai > as unless - 0 for all i not in 8.

MAJORITY GAMES AND THE MAIN SOLUTION 435

of W . In the terminology introduced in 49.6.3. : a s is required to be con-
stant for the minimal elements of W — i.e. the elements of W m .

We define accordingly:

(50:E) The weights w h • • • , w n are homogeneous, if the a s of

(50:6) have a common value, to be denoted by a, for all S of
W m .

Whenever (50 :E) is valid we shall indicate this by writing [wi, • • • , w n ]h
instead of [w h • • • , w n ].

Clearly a > 0. A common positive factor affects none of the significant
properties of w\ f • • • , w n , therefore we can use this in the case of homo-
geneity for a final normalization: Making a —

We conclude by observing that the games mentioned at the end of
50.1.3. are homogeneous and normalized by a = 1. These are the direct
majority games of an odd number of participants [1, • • • , l],.and the
corner I of Q [1,1, 1,2] — which can accordingly be written [1, • • • , 1]*
and [1, 1,1,2]*. Indeed, the reader will verify with ease that a 3 = 1 for all
S of W m in both instances.

50.3. A More Direct Use of the Concept of Imputation in Forming Solutions

60.3.1. The homogeneous case introduced above is closely connected
with the ordinary economic concept of imputation. We propose to show
this now.

More precisely: We defined in 30.1.1. a general concept of imputations
and based on it a concept of solutions. In forming these we were led by
the same principles of judgment which are used in economics, and therefore
some relationship with the ordinary economic concept of imputation must be
expected. However, our considerations took us rather far from that
concept. This applies especially to the constructions which were necessary
when we found that sets of imputations — i.e. solutions — and not single
imputations must be the subject of our theory. It will now appear that for
certain simple games the connection with the ordinary economic concept
of imputation can be established somewhat more directly. One might say
that for the special games in question the connection between this primitive
concept and our solutions can be directly established. Actually it will
provide a simple method to find a particular solution for each one of those
games.

60.3.2. The two concepts of solution, i.e. the two procedures, support
each other quite effectively. The ordinary economic concept provides a
useful surmise as to the form of a certain solution. And then our mathe-
matical theory may be used to determine the solutions in question and to
make the requirements of the ordinary approach complete. (Cf. 50.4.
on the one hand, and 50.5. et sequ. on the other.)

These considerations also serve another end: They bring out the limita-
tions of the ordinary approach with great clarity. The ordinary approach

436

SIMPLE GAMES

functions in this form only for the simple games, and even there not always
and not entirely unaided by our mathematical theory. Besides, it does not
disclose all the solutions for the games to which it applies. (Further
remarks on this subject occur throughout the discussion, and particularly
in 50.8.2.)

In this connection we emphasize again that any game is a model of a
possible social or economic organization and any solution is a possible
stable standard of behavior in it. And the games and solutions not covered
by the method referred to — i.e. by the unimproved economic concept of
imputation — will prove to be quite vital ones for social or economic theory.
It will be seen that the simple games which can be treated by this special
method are closely connected with the homogeneous weighted majority
games of which they are a generalization.

50.4. Discussion of This Direct Approach

50 . 4 . 1 . Consider a simple game T which we assume in the reduced form
with y = l, but which we do not yet restrict any further. Let us try to
discuss it in the sense of the ordinary economic ideas without making use of
our systematic theory.

Clearly, in this game the sole aim of players is to form a winning coalition,
and once a minimal coalition of this kind is formed, there is no motive for its
participants to admit additional members. Consequently one can assume
that the minimal winning coalitions — the S of W m — are the structures
that will form. It is therefore plausible to assume that a player's fate
presents only two significant alternatives: He either succeeds in joining
one of the desirable coalitions or he does not. In the latter case he is
defeated, hence he obtains the amount —1. In the former case he is suc-
cessful and according to ordinary ideas one ought to ascribe to this success
a value. This value may vary from one player to another; for player i
we denote it by — 1 + %i so that Xi is the margin between defeat and success
for player i. 1

50 . 4 . 2 . Let us now formulate the requirements which must be imposed on
these Xi, • • • , x n in the course of a conventional economic discussion.

First: By the very meaning of the Xi necessarily

(50:7) Xi ^ 0.

Second: If it happens that no minimal winning coalition contains a
certain player i, then there exists for him no alternative to the value —1,
and so we need not define any x» for him. 2

1 We assume here that there is only one way of winning, i.e. that the margin x% is the
same whichever (minimal winning) coalition the player succeeds in joining. This is
plausible since there is only one kind of success in a simple game: the complete one —
every coalition being either fully defeated or fully winning.

It will appear in 50.7.2. and 50.8.2. how far this standpoint carries. As far as it
does, it can be advantageously combined with our systematical theory.

1 For the really important simple games such i do not exist — i.e. every player belongs
to some minimal winning coalition. Cf. the first observation in 51.7.1. and (51:0)
in 51.7.3.

MAJORITY GAMES AND THE MAIN SOLUTION

437

Third: If a minimal winning coalition S becomes effective, then the
division between the players will be this : Each player i not in S obtains — 1 ,
each player i in S obtains — 1 + The sum of these amounts must be
zero. This means

o - X (-i) + X (-i +*o

x not in S x in S

i.e.

(50:8) X x ' ~ n -

t in S

In our system of notations this distribution is described by the vector

— >

a ={«!,•••, a„) with the components

( — 1 for i not in S.

7

— 1 + Xi for i in S.

We denote this vector by a s . Our first condition and the present one

actually state just that a 6 ’ is an imputation in the sense of 30.1.1.

50.4.3. Continuing the usual line of argument, we shall now want to
determine the x ly • • • , x n by means of the equations and inequalities
of the three above remarks. In doing this, one more point must be con-
sidered: We have stated in the third remark, that its S must be minimal
winning, i.e. belong to W m . However it may be asked whether all S of W m
can be used.

Indeed, the present procedure is nothing but the usual one to determine
the imputation of values to complementary goods by means of their alterna-
tive uses. 1 Now these alternative uses may be more numerous than the
different goods under consideration — i.e. W m may have more elements than n. 2
In such a situation one might expect that some of the uses are unprofitable
and need not be included in the third remark. Indeed, we already made
use of this principle by taking the S of W m only, and not all elements of W ,
because the S of W — W m (the non-minimal winning coalitions) are clearly
wasteful. Are we now sure that all S of W m must be considered as
equivalents of profitable uses? They are clearly not wasteful in the crude
sense of the S mentioned above; no participant of an S inW m can be spared
without causing defeat. But unprofitability can arise in less direct ways
than this, as numerous economic examples show. Thus the question remains
unanswered as to which S of W m are to be used in the third remark.

It is clear, however, that if an S of W m is not included there, i.e. if

(50:8) Xi = n

t in S

1 In this case it would be more suitable to say, services. The object considered is the
total service of player i in cooperating within a coalition which he joins.

* Cf. The fourth remark in 53.1.

= ~n + X x <>

i in S

438

SIMPLE GAMES

fails to hold for it, then it must be definitely unprofitable. I.e. we must
have > in place of = in (50 :8) :

(50:9) Xi > n.

i in S

Thus the question arises : By what criteria are we to determine which S
of W m fall under the third remark — i.e. for which must (50:8) hold. Denote
their set by U(s W m ). Then (50:9) must hold for the S of W m — U . So
the problem is to determine U. 1

50.5. Connection with the General Theory. Exact Formulation

50 . 5 . 1 . Instead of attempting a verbal description, let us settle this
point by going back to our systematic theory. From the statement made
in 50.4. we carry over this much: Consider a system of minimal winning
coalitions, i.e. a set U c W m and the x*. Form the imputations

as in 50.4.

{ — 1 for i not in S )
— l+Xi for i in S j

when S is in C7.

That these a 5 , S in £/, are indeed imputations, is expressed, as we know,
by the conditions of 50.4.

(50:7) x, £ 0,

(50:8) ^ Xi = n when S is in U.

x in 8

Form the set V of the a 5 , S in U. We will decide whether U and the Xi
are satisfactory, by determining whether this V is a solution in the sense
of 30.1.1.

It will be seen that the result which is obtained in this manner can be
stated verbally and is perfectly reasonable from the ordinary economic
point of view. But it may be questioned whether it could have been
unequivocally established by the usual procedures. This may serve as an
illustration of how our mathematical theory can serve as a guide even
for the purely verbal discussions of the ordinary economic approach,
(cf. 50.7.1.).

50 . 5 . 2 . We proceed to investigate whether V is a solution.

Let us determine first, when a given imputation = {Pi, • • • , p n ]
is dominated by a given a T , T in U . Since the game is simple, the set S

1 It would be utterly mistaken to try to define W m — U (and so U) by means of
(50:9). This would not restrict the Xi, • • • , x n sufficiently — and their determination
is the real objective!

MAJORITY GAMES AND THE MAIN SOLUTION 439

of 30.1.1. for this domination can be assumed to belong to W (or even to
W m , use (49:1) or (49 :J) in 49.7.2.). For every i in S, af > ft — 1;
for every i not in T, a» = - 1 : hence S £ T. Now T is in U £ W m y S is
in W, therefore S £ T yields S = T. So we see: The set S of 30.1.1. for
this domination must be our T. And T can be used there, since it is
certainly necessary, as it belongs to U £ W m £ W } cf. above. Hence the
— > — ►

domination a T H p amounts to this: af > p t for i in T , i.e.

(50:10) p t < — 1 + x x for i in T.

Denote for any imputation p = {ft, * * * , ft) the set of all i with
(50:11) ft ^ -1 +x %

by R( 0 ). Then (50:10) states that R( (3 ) and T are disjunct. An alterna-
tive way of writing this is :

(50:12) -fi(V)ar.

We repeat:

(50:F) a T & ft is equivalent to (50:12).

From this we can infer:

(50 :G) Let U* be the set of all R(£ I) which possess some subset

belonging to U.

Let U + be the set of all R(£ I) for which —R does not belong
to U*.

Then fi is undominated by any element of V if and only if
R( p ) belongs to U+.

— > — ►

Proof : That p is dominated by some element of V — i.e. by some a r ,

T in U means that (50:12) holds for some T in U. This is equivalent to

saying that — R( p ) is in U *, i.e. that R{ P ) is not in (/+.

— ►

Hence R(P) belongs to L r+ if and only if p is dominated by no
element of V.

50.5.3. Before going any further we observe four simple properties of
the set C/+ of (50 :G)

(50:H:a) U* = U+ = W if U = W"

Proof: Assume U = W m . Then U* consists of those sets which possess a
subset belonging to W m — i.e. a minimal winning subset. Hence U* = W.
The operation which leads in (50 :G) from U* to(7+is the combination of the
transformation (48:A:a) and (48:A:b) in 48.2.1. Now we noted already

440

SIMPLE GAMES

then, that these two transformations compensate each other, when applied
to W. Hence U* = W gives U + = W.

(50:H:b) U* is a monotonic and U + is an antimonotonic operation.

I.e. f/i £ Ui implies U* £ U t and Uf 2 Uf-

Proof : It suffices to recall the definitions in (50:G), to see that Ui £ U 2
implies Uf £ U * and this in turn Uf 2 Ui.

(50:H:c) All our U £ W m have U* £ W £ IJ + .

Proof: Combine (50:H:a) and (50:H:b) (with U, W m in place of Fi, U 2 ).
(50:H:d) Both U* and F+ contain all supersets of their elements.

Proof: This is obvious for U*. The property under consideration is the
same one which was formulated in (48:A:c) in 48.2.1. ( W taking the place

of our F* F+.) Now the operation which leads in (50 :G) from F* to F + ,
is the combination of the transformations (48:A:a) and (48:A:b) in 48.2.1.
(Cf. the proof of (50:H:a)). Application of (48 :B) in 48.2.2. to these two
transformations shows that the property in question is conserved when
passing from U* to U+.

60.5.4. Note that f T *, l T+ allow a simple verbal interpretation. If we
knew only of the coalitions belonging to U that they are winning, of which
coalitions could we then assert that they are certainly winning, and of
which that they are not certainly defeated?

The former is the case for the coalitions with subsets in F, i.e. for those
of U*. The certainly defeated ones are the complements of these, i.e. those
not in F+. Hence U* is the set of the first mentioned coalitions, and F+
the set of the last mentioned ones.

Now the meaning of (50:H:a)-(50:H:c) becomes clear: For U = W m ,
everything is unambiguous: The certainly winning coalitions are precisely
those which are not certainly defeated, and they form the set W. As U
decreases from W m , the gap widens. The first set decreases through subsets
of W, the second one increases through supersets of W.

The assertion of (50:H:d) is equally plausible.

60.6. Reformulation of the Result
60.6.1. (50 :G) of 50.5.2. allows us to state:

(50:1) V is a solution if and only if R(P) belongs to U + precisely

when P belongs to V.

So we must only decide when (50:1) holds. For this purpose we consider

an R in F+ and determine the fi for which R(P) = R.

Consider the three possibilities:

2 (-1) + 2 +*<) = °>

% not in R i in R <

(50:13)

MAJORITY GAMES AND THE MAIN SOLUTION

441

i.e.

(50:14)

>

£ Xi = n.

% in R <C

If a 0 with R( 0 ) = R exists, then we have

(50:15) 0 = £ 0, £ I (-1) + l (-1 + *,•),

t — 1 i not in R i in R

i.e. ^ in (50:13), (50:14). So > in (50:13), (50:14) excludes the existence

of any 0 with /?( 0 ) = R. I.e. the sets R in U+ with > in (50:13), (50:14)
need not be considered further. Consider on the other hand, an R in
with < in (50:13), (50:14). Then there are infinitely many ways of choos-

for i not in R
+ Xi for i in R j

— > _ i

ing 0 with £ fa = 0 and 0* ^

»-i

For all these

R( 0 ) necessarily 2 R. Hence it belongs to V by (50:H:d).

Since V is
I.e. sets

finite, these 0 cannot all belong to V. This is a contradiction.

R in U + with < in (50:13), (50:14) must not exist.

50.6.2. It remains for us to consider the sets which are in [/+ with = in
(50:13), (50:14). According to the above, these must furnish precisely

the 0 of V.

If 0 belongs to V, i.e. 0 = a T , T in [/, then we have this situation:

R ( 0 ) is T plus the set of those i for which x* = 0. T belongs to U Q U* c [/+

(for the second relation use (50:H:c)), hence R(P) belongs to U+.
Also

l l Xi = n ■

% in R(fi) % in T

So we have = in (50:13), (50:14). Hence the 0 of V are all taken care of.

Conversely: Consider an R in U+ with = in (50:13), (50:14). Addition
of all i for which Xi = 0 to R affects neither the fact that R belongs to [/+
(by (50:H:d)), nor the equation (50:14). So we may assume that R con-
tains all these i.

— > — ►

If now an imputation 0 has R( 0 ) = R } then 0< ^ — 1 + Xi for i in R .

n

Always 2: — 1. As £ /3< = 0, this implies:

•-1

Pi =

~l+Xi

for i not in R,
for i in R.

(50:16)

442

SIMPLE GAMES

Conversely: (50:16) implies that P is an imputation with R( p ) = ft.

Hence our requirement in this case must be that the p of (50:16) be an

a r , T in U . This means, that T and R differ only in elements i for which
Xi = 0. And this property is insensible to our original modification of R ,
the inclusion of all such i into ft.

Summing up:

(50 :J) V is a solution if and only if this is the case: Call an i indiffer-

ent when x x = 0. 1

Then we have

(50:8*) £ = n

x in T

for the T of U and, of course, also for those which differ from
these only by indifferent elements.

And we must require

(50:9*) £ x x > n

% in T

for all other T of U + .

In making use of this result, one may chose the set U s W m first, then
attempt to determine the a ;< from (50:8*) and finally verify whether these
fulfill the inequalities

(50:7) Xi Z 0

and (50:9*).

50.7. Interpretation of the Result

60 . 7 . 1 . The result (50 :J) permits the verbal statement promised in
50.5.1. This is it:

A solution V is found by choosing arbitrarily the set U of those minimal
winning coalitions (i.e. U c W m ), which are to be considered profitable.
The Xi must then satisfy the corresponding equations (50:8*). But after
this, it must be verified that certain other coalitions are definitely inprofit-
able in the sense of (50:9*). This must be required not only for those coali-
tions which are known to be winning, (i.e. W ), but for all those which cannot

1 These i constitute a slight complication which is further aggravated by the fact
that we have no example of a game in which they actually occur. It may be that they
never exist; an indifferent i characterizes a player who belongs to some minimal winning
coalitions, but never receives a share.

The excluded player in a discriminatory solution of the three-person game is in this
situation (cf. (32: A) in 32.2.3. with c — 1). But that solution is an infinite set, whereas
our V must be finite.

It would be of interest to decide this existential question. At any rate we must at
present provide for the indifferent i to avoid loss of generality or rigour.

MAJORITY GAMES AND THE MAIN SOLUTION 443

be established as definitely defeated by the coalitions of U alone (i.e. C7+) —
excepting, of course, the coalitions of U itself. 1

The reader may now judge whether the concluding remark of 50.5.1.
is justified by this formulation.

50 . 7 . 2 . The question of finding the proper U for (50:1) is a rather deli-
cate one. The antimonotony of U + (cf. (50:H:b) in 50.5.3.) makes itself
felt now: Decreasing 17, i.e. the number of equations, increases U + , the
number of inequalities, and vice versa.

In particular, if we choose U as large as possible, i.e. U = W m } then the
inequalities associated with U + create no difficulties at all. Indeed:
U = W m implies U + = W by (50:H:a) in 50.5.3. A T of W certainly
possesses a subset S which is minimal in W , i.e. belongs to t7 = W m . Now
if T differs from this S by more than indifferent elements, then we have
Xi > 0 for some i in T — 5, hence x t > £ Xi = n, i.e. (50 :9*) as desired.

» in T t in 8

Thus U = W m always yields a solution V, if its equations (50:8*) can
be solved at all (with (50:7)).

But as we pointed out in 50.4.3., we have no right to expect a priori
that this will always be the case — especially since there may be more equa-
tions (50:8*) (i.e. elements in W m ), than variables Xi.

The last objection is not an absolute one; indeed it is easy to find a
simple game for which the number of these equations exceeds the number
of variables and the solution nevertheless exists. 2 On the other hand
there exist simple games for which those equations have no solutions. An
example of this is somewhat more hidden, 8 but the phenomenon is probably
fairly general. When this occurs, one must investigate whether a solution
V cannot be found by appropriate choices U c W m . The difficulty and
delicacy of this question has been commented upon already at the beginning
of this section. 4

50.8. Connection with the Homogeneous Majority Games

60 . 8 . 1 . We now restrict ourselves to the case U = W m . I.e. we assume
that the full system of equations

(50:17) £ Xi = n for all S in W m ,

«in£

can be solved with

(50:7) ^ 0.

1 And those which differ from them only by indifferent elements.

1 This happens for the first time for n -» 5, cf. the fifth remark in 53.1.

1 This happens for the first time for n *» 6, cf. the fifth remark in 53.2.5.

4 No instance of a simple game with a solution V derived from U c W m is known, nor
is it established that none exists. The further-going question whether every simple
game possesses solutions V of suitable U £ W m is equally open.

The problem seems to be of some importance. It may be difficult to solve it. It
appears to have some similarity with the solved questions mentioned in footnote 1 on
p. 154, but it has not been possible, so far, to exploit this connection.

444

SIMPLE GAMES

We saw that in this case the set V of all a s , S in W m , is a solution. In this
situation and only then, we call V a main simple solution of the game.

There is a certain similarity between these requirements and those
which characterize a homogeneous weighted majority game. Indeed, the
latter are defined by

(50:18) £ Wi = b for all S in W m

i in S

where

n

b = i ( 2) v>i + «)> a > 0 (combine (50 :D) (50 :E) of 50.2.)

»-i

and

(50:19) ^ 0.

Actually, there is more than similarity. Thus, if a system of fulfilling
(50:18), (50:19) is given, a system x< fulfilling (50:17), (50:7) obtains as
follows: The quantity b of (50:18) is positive. 1 Multiplication of all Wi
by a common positive factor leaves everything unaffected, and by choosing
this factor as n/b we can replace b in (50:18) by n. Now we can simply put
Xi ss Wi and (50:18), (50:19) become (50:17), (50:7).

If conversely a system of Xi fulfilling (50:17), (50:7) is given, there is an
extra difficulty. We may put Wi m x*. 2 Then (50:7) becomes (50:19)

n

and (50:17) yields (50:18) with b = n, i.e. a = 2n — 2} But now the

»-i

question arises whether the last requirement a > 0 is fulfilled — i.e. whether

n

(60:20) £ < 2 n.

»- 1

Summing up:

(50 :K) Every homogeneous, weighted majority game possesses a

main simple solution.

Conversely, if a (simple) game possesses a main simple solu-
tion, homogeneous weights for the game can be derived from it
if and only if (50:20) is fulfilled.

50 . 8 . 2 . This connection between homogeneous weights and main simple
solutions is significant. But it must be stressed that a homogeneous,
weighted majority game will in general have other solutions besides the

1 Otherwise all i occurring in the S of W m would have w% — 0 by (50:18) and (50:19).
Then (50:6) of 50.2.1. and (50:19) gives a* ^ 0 for the S of W m , hence a 0, which is
not the case.

1 The i which belong to no minimal winning set cause a slight disturbance, since they
have no Xi (cf. the second remark in 50.4.), while we require their ie<. However, the
contingency is unimportant (cf. loc. cit.) and we can put these u>, — 0 as is easily con-
cluded from the references of footnote 2 on p. 436.

ENUMERATION OF ALL SIMPLE GAMES 445

main simple one. 1 And a game with a main simple solution may not fulfill
(50:20), i.e. there need not be < in

TV <

(50:21) X = 2n. s

<-i >

Beyond all this, finally, we must not lose sight of the main limitation
of these considerations: Whether we take the concept of “ordinary”
imputation in its narrower form of 50.8.1. (i.e. V = W m ) or in its wider
original form of 50.6., 50.7.1. (i.e. U z W m , cf. (50:1) in 50.6.2.), it is cer-
tainly restricted to simple games. That it is necessary to go beyond these,
and beyond the special solutions described here, and that this forces us to
fall back completely on the systematical theory of 30.1.1., was pointed out
at the end of 50.3.

61. Methods for the Enumeration of All Simple Games
51.1. Preliminary Remarks

61.1.1. Beginning with 50.1.1. we introduced specific simple games which
permitted characterization by numerical criteria instead of the original
set theoretical ones (cf. the beginning of 50.2.1.). We saw, however, that
these numerical procedures could be carried out in several ways and that
there was no certainty that all simple games could be accounted for with
their help. It is therefore desirable to devise combinatorial (set theoretical)
methods that produce systematic enumeration of all simple games.

This is, indeed, indispensable in order to gain an insight into the possi-
bilities of simple games and particularly to see how far the above mentioned
numerical procedures carry us. It will appear that the decisive examples
of the non-obvious possibilities obtain only for relatively high numbers of
players,* so that a mere verbal analysis cannot be very effective.

61.1.2. We pointed out at the end of 49.6.3. that the enumeration of all
simple games is equivalent to the enumeration of their sets W, i.e. of all
sets W which fulfill (49 :W*) in 49.6.2. We also noted there that it may be
advantageous to replace the use of W (all winning coalitions), by W m
(all minimal winning coalitions).

Either procedure provides an enumeration of all simple games. The
use of W is preferable from the conceptual standpoint since W has the

1 The main simple solution of the essential three-person game ([1,1,1]*, cf. the end of
50.2.) is the original solution of 29.1.2., i.e. (32 :B) of 32.2.3. We know from 32.2.3. and
33.1. that other solutions exist.

The main simple solution of corner I of Q ([1,1, 1,2]*, cf. the end of 50.2.) is the
original solution of 35.1.3. We will discuss this game, together with the more general
one [1, • • • , 1, n — 2]* (n participants) in 55. and obtain all solutions.

All these references make it clear that the solutions other than the main simple one
are quite significant, cf. 33.1 and 54.1.

* « occurs for the first time for n e 6, cf. the fourth remark of 53.2.4. > occurs for
the first time for n ■» 6 or 7, cf. the sixth remark of 53.2.6.

Both these examples are quite interesting in their own right.

* n - 6, 7 cf. 53.2.

446

SIMPLE GAMES

simpler definition and W m was introduced indirectly with the help of W.
For a practical enumeration of all simple games — which is our present aim —
the use of W m is preferable since W m is a smaller set than W 1 and therefore
more readily described.

We will give both procedures successively. It will appear that these
discussions provide a natural application of the concepts of satisfactoriness
and saturation introduced in 30.3.

61.2. The Saturation Method : Enumeration by Means of W

61 . 2 . 1 . The sets W are characterized by (49 :W*) in 49.6.2. i.e. by the
conditions (49:W*:a)-(49:W*:c) which constitute (49:W*).

Let us for a moment disregard (49:W*:c), and consider (49:W*:a),
(49:W*:b). These two conditions imply that no two elements of W can
be disjunct. 2 In other words: Denote the negation of disjunctness — i.e.
of Sc\Tj*Q — by S(RiT. Then (49:W*:a), (49:W*:b) imply (Ri-satis-
factoriness. 3 A more exhaustive statement along these lines is this:

(51 :A) (49:W*:a), (49:W*:b) are equivalent to (Ri-saturation. 3

Proof : (Ri-saturation of W means this:

(51 :1) S belongs to W, if and only if S n T ^ © for all T of W.

(49:W*:a), (49:W*:b) imply (51:1): Let W fulfill (49:W*:a), (49:W*:b).
If S belongs to W y then we know that S n T © for all T of W. If S
does not belong to W , then T = — S belongs to W by (49:W*:a), and
5 n T = ©.

(51:1) implies (49:W*:a), (49:W*:b): Let W fulfill (51:1). We prove
(49:W*:a), (49:W*:b) in the reverse order.

Ad (49:W*:b): If S meets the criterion of (51:1), then every superset
of S does too. Hence W contains the supersets of its elements.

Ad (49:W*:a): Owing to the above, — S is not in W if and only if no
subset of —S is in W. I.e. when every T of W is not £ — S, or again, when
for every T of W, S n T ^ ©. By (51:1) this means precisely that S is
in W.

Thus, at any rate, precisely one of S, — S belongs to W.

Now jS<RiT is clearly symmetric, hence we can apply (30 :G) in 30.3.5. 4

1 W, L are disjunct sets. They have the same number of elements owing to (48:A:b)
in 48.2.1. Together they exhaust I which has 2 n elements. Hence W as well as L has
exactly 2 W ~ 1 elements.

The number of elements in W m varies, but it is always considerably smaller. (Cf.
the fourth remark in 53.1.)

* Proof: Let S, T belong to W, S n T — ©. Then hence —5 belongs to IF by

(49:W*:b), thus violating (49:W*:a).

1 Cf. the definitions of 30.3.2.

4 It will be remembered that we also assumed in 30.3.5. the general validity of x(Rx —
i.e. in this case of This means S & © — po it fails for S ■■ ©.

However, (49:W*:a), (49:W*:c) exclude © form W t hence we may use as domain D
in the sense of 30.3.2. instead of / (the system of all subsets of I) equally well I — (©) (the

system of all non-empty subsets of I). This rids us of S - ©.

ENUMERATION OF ALL SIMPLE GAMES

447

51 . 2 . 2 . In order to discuss (49 :W*) on this basis, we must take (49:W*:c)
also into account. This can be done in two ways. The first way will be
useful for a subsequent comparison.

(51 :B) W fulfills (49 :W*) if and only if it is (Ri-saturated and con-

tains neither © nor any one-element set.

Proof: (49:W*) is the conjunction of (49:W*:a), (49:W*:b) and
(49:W*:c). The first two amount by (51 :A) to (Ri-saturation. Taking
(49:W*:a) for granted, (49:W*:c) may be stated thus: If S is I or an (n — 1)-
element set, then — S is not in W. I.e.: Neither © nor any one-element
set is in W.

The second way is more directly useful.

Let Vo be the system of all sets of (49:W*:c) — i.e. of I and all (n — 1)-
element subsets of /. Then we have:

(51 :C) V is a subset of a IF fulfilling (49:W*) if and only if V u Vo

is (Ri-satisfactory.

Proof: W 2. V and W fulfilling (49 :W*) amount to this: W 2 V, W ful-
fills (49:W*:a), (49:W*:b)-~ i.e. W is (Rrsaturated by (51 :A) — W fulfills
(49:W*:c) — i.e. W 2 V Q . In other words: We are lookingfor an (Ri-saturated
W 2. V u Vo — i.e. we are asking whether 7u7 0 can be extended to an
(Ri-saturated set.

Now we know that (30 :G) of 30.3.5. applies, and hence the considera-
tions of the last part of 30.3.5. apply too. 1 This extensability is equivalent
to the (Ri-satisfactoriness of V u F 0 .

51 . 2 . 3 . We rephrase (51 :C) more explicitly :

(51 :D) V is a subset of a IF fulfilling (49 :W*) if and only if it pos-

sesses these properties:

(51:D:a) No two S, T of V are disjunct.

(51 :D :b) V contains neither © 2 nor any one-element set.

Proof: We must express, according to (51 :C), the (Ri-satisfactoriness of
V vV 0 . I.e. that no two S, T of V or Vo are disjunct.

S, T are both in V: This coincides with (51:D:a).

S y T are both in W* Both have ^ n — 1 elements, hence they cannot
be disjunct. 3

Of Sy T one is in V and the other in W We may assume by symmetry,
S as the former and T as the latter. So an S of V must not be disjunct with
I or any (n — l)-element set. This is precisely (51 :D:b).

1 Note that the domain D — / — (©) (cf. footnote 4 on p. 446) is finite.

* For this cf. also footnote 4 on p. 446.

* We are using that 2 (n — 1) > n i.e. n > 2 i.e. n ^ 3. This should have been stated
explicitly at the beginning — but it is a natural assumption, since simple games (i.e. sets
with (49:W*)) exist only for n £ 3. (Cf. 49:4, 49:5.)

448

SIMPLE GAMES

(51 :D) solves the question of enumerating all W : Starting with any V
which fulfills (51:D:a), (51 rDib) 1 we may increase it gradually as long as
this can be done without violating (51:D:a), (51:D:b). When this process
cannot be continued any further, than we have a V which is maximal
among the subsets of the W (with (49 :W*)) — i.e. we then have such a W.

In performing this gradual building up process in all possible ways, we
obtain all W in question.

The reader may try this for n = 3 or n = 4. It will appear that the
procedure is quite cumbersome even for small n, although it is rigorous and
exhaustive for all n.

51.3. Reasons for Passing from W to W m . Difficulties of Using W m

51 . 3 . 1 . Let us consider the sets W m of 49.6.

We wish to characterize these W m directly and to find some simple process
to construct them all. In what follows, we will derive two different ways
of characterization, both being of the saturation type. The first will be by
means of an asymmetric relation, while the second will be by a symmetric
one. Thus it is the second one which is suited for construction purposes, in
analogy with the construction of W in 51.2.

We give nevertheless both characterizations because the equivalence is
quite instructive: The first one is in some (technical) respects similar to the
definition of a solution (cf. 30.3.3. and 30.3.7.), and therefore the transition
to the equivalent second form is of interest since it points a way to solve
problems of this type. We have mentioned before (in 30.3.7.) how desirable
the corresponding transition for our concept of solution would be.

51 . 3 . 2 . Let W be a system which contains all supersets of its elements:
e.g. fulfilling (49 : W * :b) . Then the system of its minimal elements W m deter-
mines W : Indeed, it is clear that W is the system of the supersets of all
elements of W m .

Hence if a system V is given, and we are looking for a W with (49 :W*)
such that V = W m , then this W must necessarily be the system V of the
supersets of all elements of V.

Consequently V = W m for a W with (49 :W*) if and only if these two
requirements are met by W = V. 2 We are now going to transform this
characterization of the V = W m into one of the saturation type.

Denote the assertion that neither S n T = © nor Sd T, by &<R%T.
Then we have:

(51 :E) V = W m for a W with (49 :W*) if and only if V is (Ri-saturated

and contains neither © nor any one-element set.

Proof: According to the above, we must only investigate whether W = V
has the desired properties:

x In principle we may start with the empty set. The reader will note that the exclu-
sion of © from V (cf. above) does not affect the possibility of V * ©.

* I.e. W ■■ V is the only system which can possibly meet these requirements, but even
it may fail.

ENUMERATION OF ALL SIMPLE GAMES 449

V as W m : Let S be a minimal element of this W. Then S 2 . T for some
T of V. Hence T is in W, and so the minimality of S excludes S d T. So
S = T i.e. S belongs to V.

Thus only the converse property must be discussed: Whether every
S of V is really minimal in W. Any S of V clearly belongs to W. So the
minimality means that 8 d T' y T' of W is impossible; i.e. that S d T' 2 T,
T of V is impossible. This implies the impossibility of S d T y T of V and
is implied by it (put V = T). So we have this condition:

(51 :2) Never S d T for S y T in V.

W fulfills (49 :W*): We must consider (49:W*:a), (49:W*:b), (49:W*:c)
separately. We do this in a different order.

Ad (49:W*:b): Clearly W = V contains all supersets of its elements so
this is automatically fulfilled.

Ad (49:W*:c): Take (49:W*:a) for granted. (Cf. below.) Then
(49:W*:c) may be stated thus: If S is I or a (n — l)-element set, then — S is
not in W. I.e. neither @ nor any one-element set is in W; that is, no subset
of these is in V. So we have this condition:

(51:3) Neither Q nor any one-element set is in V .

Ad (49:W*:a): We consider this in two parts:

S' y — S' cannot both belong to W: I.e. if S y T belong to V y then we can-
not have SsS' y —S'. Now the existence of such an S' implies
S n T = © and is implied by it (put S' = S). So we have this condition:

(51:4) Never S n T = 0 for S y T in V.

One of S, —S must belong to W: Assume that neither of S y — S belongs
to W. This means that no T of V has T £ S or T £ —S y the latter meaning
S n T = ©. I.e. no T of V has T = S or £ d T or S n T = ©. Or again:
S is not in V and no T of V fulfills the negation of S^T. 1
I.e. S is not in V, but S6iaT for all T of V.

Now we have to express that this is impossible: i.e.:

(51:5) If SGiaT for all T of V y then S belongs to V.

Thus (51:2)-(51:5) are the criteria we want.

Now (51:2) and (51:4) can be stated together like this: SGiaT for all S y
ToiV . I.e.:

(51:6) SfaaT for all T of V, if S belongs to V .

(51:5) and (51:6) together express precisely the (R r saturation of V ,
Hence this and (51:3) form the criterion — and this is precisely what we
wanted to prove.

(51 :E) is of some interest because it is a perfect analogue of (51 :B).
Thus these characterizations of W and W m differ only in the replacement of

1 This is indeed SoTorSnT-Q.

450

SIMPLE GAMES

S(RiT: not S n T = ©

by

S(R 2 T : neither S n T = © nor SdT.

But as this replaces the symmetric (Ri by the asymmetric (Rj, (51 :E) cannot
be used in the way in which we used (51 :B) — or rather the underlying
(51 :A).

61.4. Changed Approach : Enumeration by Means of W m

51 . 4 . 1 . We now turn to the second procedure. This consists in analyz-
ing the following question : Given a system V, what does it mean for a W
fulfilling (49 :W*) that Vs W m ?

The meaning of the V s W m is this : Every S of V is a minimal element
of W. I.e. such an S must belong to W but its proper subsets must not
belong to W. As W fulfills (49:W*:b), i.e. contains the supersets of all its
elements, it suffices to state this for the maximal proper subsets of S only;
i.e. for the S — (i), i in S. As W fulfills (49:W*:a) we may say instead of
S — (t) not belonging to IV, that — (S — (i)) = (—5) u (i) is in W. So
we see:

(51 :F) V £ W m (W with (49:W*)) means precisely this: For every

S of V, S belongs to W; and for every i of this S, (-S) u (i)
belongs to W.

We now prove:

(:51:G) V is a subset of the W m of a W fulfilling (49:W J, ‘) if and

only if it possesses these properties:

(51:G:a) No two S, T of V are disjunct.

(51:G:b) No two S, T of V have Sd T.

(51:G:c) For S, T of V, S u T = / implies that S n T is a one-

element set.

(51:G:d) Neither © nor any one-element set, nor I must belong

to V.

Proof: Let V\ be the set of all (-S)u (i), S in V , i in S. Then V £ W m
means by (51 :F), that Vo V\S W. This is possible for some W with
(49 :W*) according to (51 :D), if V u V x fulfills (51:D:a), (51:D:b).

Let us therefore formulate (51:D:a), (51:D:b) for V u V\.

Ad (51 :D:a) : S , T are both in V : This coincides with (51 :G:a).

S, T are both in Vi: I.e. S = ( — S') u (i), T = (-T')o (j), S', T in V, i in
S', j in T .

The disjunctness of S , T means these: —5', — T' disjunct, i.e. S' u T = 7;
(i), (J) disjunct, i.e. i j* j ; -S', (j) disjunct, i.e. j in S'; - T', ( i ) disjunct,
i.e. i in T'.

Summing up: S' u T' = I; i , j two different elements of both S' and T' —
i.e. of S' n T'.

ENUMERATION OF ALL SIMPLE GAMES 451

Now we must state that this is impossible. I.e. if S' u f = /, then
S' n T 9 cannot possess two different elements. As S' n T' cannot be empty
by (51:G:a), this means that it must be a one-element set.

Thus precisely (51 :G:c) obtains (S', T' in place of its S, T).

Of S, T one is in F and the other in Fr. We may assume by symmetry,
that S is the former and T the latter. So T = ( — T') u (j), T 9 in F, j
in T 9 . The disjunctness S, ( — T 9 ) u (j) means: S, — T 9 are disjunct, i.e.
SsP; S, (j) are disjunct, i.e. j not in S.

Summing up: S £ T', j an element of T' not in S.

Now we must state that this is impossible. I.e. not S c T 9 . Thus
precisely (51:G:b) obtains. (7 1 ', S in place of its S, T .)

Ad (51:D:b): Neither © nor any one-element set must belong to F nor
to V The latter means that neither must be a ( — S) u (i), S in V, i in S.
Only a one-element set could be such a (— S) u (i) and this would mean:
— S ~ ©> i e. S = J.

Summing up: Neither © nor any one-element set, nor I must belong
to F. This coincides with (51:G:d).

Thus we have obtained precisely the conditions (51 :G:a)-(51 :G:d) as
desired.

(51 :G) solves the problem of enumerating all W m in perfect analogy
to the solution by (51 :D) of the corresponding problem for the W : Starting
with any F which fulfills (51 :G:a)-(51 :G:d) 1 we increase it gradually as
long as this can be done without violating (51 :G:a)-(51 :G:d). When this
process cannot be continued any further, we have a F which is maximal
among the subsets of the W m of a IF with (49 :W*) — i.e. we have such a W m .

In performing such a gradual process of building up in all possible ways,
we obtain all W m in question.

61 . 4 . 2 . Our last remarks show that the practical enumeration of all
simple games can be based on (51 :G) — and we will, indeed, undertake it in
52. But some other considerations are better carried out at first.

We now propose to analyze the assertion that (51 :G) is a condition of the
saturation type a little more closely.

Observe first, that as (51:G:b) refers to two arbitrary S, T of F, we
can interchange these in it. I.e. we can replace (51:G:b) by this:

(51 :G:b*) No two S, T of F have S d T or S c T.

Denote the assertion that S f T fulfill (51:G:a), (51:G:b*), (51:G:c) —
i.e. that neither S n T — © nor S d T 7 , nor ScT, nor S u T = / without
S n T being a one-element set — by

Then (51 :G) simply states that F is (Rj-satisfactory, together with
(51:G:d). Now let the domain D be the system 7 of those subsets of I
which fulfill (51:G:d) — i.e. neither ©, nor a one-element set, nor /. Then
the last remarks of 51.4.1. show that the W are the maximal (Rs-satisfactorv
subsets of 7 .

1 In principle we may start with the empty set.

452

SIMPLE GAMES

S(RzT is clearly symmetric. 1 Hence we may apply (30 :G) of 30.3.5.
This gives:

(51 :H) V = W m for a W with (49 : W*) if and only if V is (R 3 -saturated

(in I).

Comparing (51 :E) with (51 :H) shows that we have succeeded in passing
from the asymmetric (Ra to the symmetric (R 8 — fulfilling the promise made in
footnote 1 p. 271.

51 . 4 . 3 . It is quite instructive to compare (Ha (in 51.3.2.) with our (H 8 :

S(R 2 T: neither S n T = ©, nor S o T.

S(R 3 T: neither S n T = ©, nor Sz> T, nor S cT,

nor S u T = I without S n T being a one-element
set.

Mere symmetrization of (R 2 (cf. 30.3.2.) would give the three first parts
of (R 8 , but not the last one. This last part is the essential achievement of
(51 :G) and (51 :H) and not connected in any obvious way with the three
others.

One can infer from this how recondite the operations must be by which
the program of 30.3.7. might be carried out — if this proves to be feasible
at all.

61.6. Simplicity and Decomposition

51 . 5 . 1 . Let us consider the connections between the concept of a simple
game and that of decomposition.

Assume, therefore, that T is a decomposable game with the constituents
A, H ( J , K complements in I). Then we must answer this question:
What does it mean for A, H that T is simple?

We begin by determining the sets W> L. Since we must consider them
for all three games T, A, H, it is necessary to indicate this dependence.
We write therefore Wr, Lr ; W a, La; Wh, Lr.

It should be added that we assume neither essentiality nor any normal-
ization for the games T, A, H. It is convenient, however, to assume them
all in a zero-sum form. 2

(51:1) S = RuT(R£j, T £ K) belongs to W T [Lr] if and only if

R belongs to W& [La] and T to W H [Lr].

Proof: Replacement of S by its complement (in /), / — S, 3 replaces R ,
T by their respective complements (in L, K). This transformation inter-

1 And £<Rs/S holds in I:S n S ~ © occurs only for S - 0, S => S never, S U S = /
only for S — I — hence neither of these happen for an S of 7.

* The reader who recalls the discussions of 46.10. may want to know at this point how

the question of the excesses (in T, A, H the e<>, ^ loc. cit.) is to be handled. This ques-

tion will be clarified in the discussion of 61.6.

* It is preferable to write the complement in this way, instead of the usual — <S>,

— T y since we are complementing in different sets.

ENUMERATION OF ALL SIMPLE GAMES

453

changes Wr, Wa, W H with L r, La, L h . Hence our statement concerning
the W implies that one concerning the L and vice versa. We are going to
prove the latter.

That S belongs to Lr is expressed by
(51:7) v(S) = l v((t))

t in S

since A, H are the constituents of r, we have v(S) = v(f?) + v(T). Hence
we can write (51:7) thus:

(51:8) v(R) + v(T) = £ v((t)) + £ v((*)).

i in A t in T

That ft belongs to La and T to L H is expressed by

(51:9) v(«) = X v («).

i in R

(51:10) v(T) = X v («)-

t in T

The assertion which we must prove, then, is the equivalence of (51 :7) and
(51:9), (51:10).

Clearly (51:9), (51:10) imply (51:7); the reverse implication can be
drawn since always

v(R) ^ X v (( z '))>

i in R

y(T) ^ X v («)

* in T

(cf. (31:2) in 31.1.4.).

51 . 5 . 2 . We are now able to prove:

(51 :J) T is simple if and only if this is true: Of the two constituents

A, H one is simple, and the other is inessential.

Proof: The condition is necessary: Simplicity of T means this:

(51:11) For any S £ I one and only one of these two statements is

true:

(51:ll:a) S is in Wr.

(51 :11 :b) S is in L r .

Put S = flu T (Bs J } TzK) and apply (51:1) to (51:11). Then this
results:

(51^2) For any two R £ J, T £ K one and only one of these two

statements is true:

(51:12:a) R is in Wa and T is in Wh.

(51 :12:b) R is in La and T is in Lh.

454

SIMPLE GAMES

Now put R = ©, T = K. Then R belongs to L A and *T belongs to Wr.
Hence for (51 :12:a) W a and L A have a common element: R } and for (51 :12:b)
Wn and L h have a common element: T. By (49 :E) in 49.3.3. (applied to
A, H instead of its r) the former implies that A is inessential, and the
latter, that H is inessential.

So we see:

(51:13) If T is simple, then either A or H is inessential.

The condition is sufficient: We assume, by symmetry, that H is inessen-
tial. Then (49 :E) in 49.3.3. (applied to H in place of its T) shows that every
T S K belongs to both Wr and L H . Hence we can now reformulate the
characterization (51:12) of the simplicity of T.

(51:14) For any R s J one and only one of these two statements is

true:

(51:14:a) R is in W A .

(51 :14:b) R is in L A .

This is precisely the statement of the simplicity of A. So we see:

(51:15) If H [A] is inessential, then the simplicity of T is equivalent

to that one of A [H].

(51:13), (51:15) together complete the proof.

61.6. Inessentiality, Simplicity and Composition. Treatment of the Excess

61 . 6 . It is worth while to compare (51 :J) with (46:A:c) of 46.1.1.
There we found that a decomposable game is inessential if and only if
its two constituents are — i.e. inessentiality is hereditary under composition.
This is not true for simplicity, which as we know, is the simplest form of
essentiality: By (51 :J) a decomposable game is not simple if its two con-
stituents are. (51 :J) shows that a simple game A remains simple under
composition if and only if it is combined with an inessential game H — i.e.
with a set of “dummies” (cf. footnote 1 on p. 340).

In this connection four further remarks are appropriate:

First: If the simple game T obtains as described above from the con-
stituent (simple) game A by an addition of “dummies” (i.e. of the inessen-
tial game H), then the solutions of T are directly obtainable from those of A.
Indeed, this is described in detail in 46.9. 1

Second: We stated at the beginning of 49.7. that we use the old form of
the theory for simple games. It is therefore worth noting that the type of
composition to which we were led (cf. the above remark) is precisely the
one for which the old form of the theory is hereditary. (Cf. the end of
46.9. or (46 :M) in the first remark in 46.10.4.)

1 This is, of course, what common sense leads one to expect anyhow. The surprising
turns of the theory of decomposition — cf. in particular the resum6 in 46.11. — show,
however, that it is unsafe to lose sight of the exact results. In this case 46.9. provides the
firm ground.

ENUMERATION OF ALL SIMPLE GAMES

455

Third: In this connection it becomes clearer also why we had to refrain
from considering other excesses than zero — i.e. the new form of the theory
in the sense of 44.7. — for the theory of simple games.

Indeed: If we had been able to carry this out successfully, then the
results of 46.6. and 46.8. would enable us to deal with all compositions of
simple games. Now we have seen that a composition of simple games is not
a simple game. In other words: A theory of simple games with general
excess would indirectly embrace non-simple games as well. It is therefore
not surprising that we could not proceed in generality. 1

Fourth: In the light of the analysis of 46.10. the above remarks concern-
ing the excess assume the following significance: They show that the con-
cept of simplicity does not stand the general operation of imbedding. 2
This shows that the methodical principle considered in 46.10.5. cannot be
applied under all conditions.

51.7. A Criterion of Decomposability in Terms of W m

51 . 7 . 1 . In 51.5. we discussed when a decomposable game T is simple.
We now tackle the converse problem: To decide when a simple game is
decomposable.

Let a simple game T be given. It will appear that the following concept
is of importance: An i of 7 is significant if and only if it belongs to some S
of W m . z Denote the set of all significant elements of 7 — i.e. the sum of all
S in W m — by 7 0 .

We now proceed in several successive steps:

(51 :K) If T is simple and decomposable, and if the simple constituent

is A (cf. (51 :J) and the use of notations of 51.5.) then T and A
have the same W m .

Proof: According to (51:1) the S = R u T (RzJ, T £ K) of Wr
obtain by taking any R of Wa and any T of W K . H is inessential (by
(51 :J)), hence the T of Wn are simply all T £ K (cf. the proof of (51 :J)).
Consequently this S = R u T is minimal — i.e. it belongs to W ™ — if its
R, T are minimal. This means that R belongs to Wj and that T = ©, i.e.
S = R.

Thus If" and Wj coincide, i.e. T and A have the same W m .

(51 :L) With the same assumptions as in (51 :K), necessarily J 2 / 0 .

Proof: T and A have the same W m (by (51 :K)), hence the same significant
elements — therefore those of T y which form the set 7 0 , are all among the par-
ticipants of A, which form the set J.

1 In a certain sense this may be viewed as an application of the methodical principle
referred to in footnote 3 on p. 270.

1 Unless it is merely an addition of “dummies” as discussed above.

* Thus a player i is significant if there exists a minimal winning coalition to which he
belongs; i.e. if there exists a conceivable essential service he may render.

It will be seen that the opposite of this is a “dummy ” (cf. the end of 51.7.3.).

All this refers, of course, to simple games.

456

(51 :M)

SIMPLE GAMES

Assume only that F is simple. Then Jo is a splitting set, 1
the /o-constituent A being simple, and the (J — J 0 )-constituent
H inessential (cf. (51 :J)).

Proof: Consider an S = R u T, R £ J 0 , T £ I — Jo. Then:

(51:16) S is in W if and only if R is in W.

Indeed: If R is in W, then S 2. R is too. Conversely: Let <S be in W.
Then a minimal T in W with T £ S exists. So T is in W m , every i of T is
in J 0 . Hence T £ 7 0 . Thus TsSnh — R, and therefore R is in W
along with T.

(51:17) TisinL.

Indeed: Replace S by T(s I — Jo); this replaces our R, T by @, T.
As © is in L, (51 :16) permits to infer the same for T.

We now prove:

(51:18) v(S) = v(R) + v(T).

Consider the S of L and of W separately :

S is in L: R, T £ S are also in L. Hence

V(S) = X v((t')) = X v((0) + 2 v («) =

i in S i in R i in T

i.e. (51:18).

S in W: By (51 :16), (51 :17) R is in W and T in L. Hence

vos) = - 2 v (W)>

i not in S

v(«) = - 2 = - 2 v (W) - 2 y (W)>

i not in R * not in 3 t in T

v(T) = 2 V (W),

* in T

and so

v(S) = v(R) + v(T).
i.e. (51:18)

(51:18) is precisely the statement that Jo is a splitting set. For all
TsI — Jo (51:17) gives

v(T) = 2

* in T

hence the / — / 0 constituent H is inessential. Consequently the Jo-con-
stituent A must be simple by (51 :J).

Thus the proof is completed.

] In the sense of 43.1.

SIMPLE GAMES FOR SMALL n

457

51.7.2. We are now able to describe the decomposibility of a simple
game T completely — i.e. we can name its decomposition partition Ilr in the
sense of 43.3.

(51 :N) With the same assumptions as in (51 :M) : The decomposition

partition n r consists of the set 7 0 and of the one-element sets
(i) for all i in 7 — 7 0 .

Proof : All (i), i in 7 — 7 0 , belong to II r : By (51 :M) 7 — 7 0 is a splitting
set of T with an inessential constituent H. Hence every (i), i in 7 — 7o, is a
splitting set of H (use, e.g. (43 :J) in 43.4.1.) and so of T (use (43 :D) in
43.3.1.). Being a one-element set, ( i ) is necessarily minimal. Hence it
belongs to II r .

7o belongs to Ilr: 7 0 is a splitting set by (51 :M). If / is a splitting set
9* ©, then (51 :L) applies to J or to 7 — J, hence either /27 0 or7 — / 27 0 ,
7 0 n / = ©. Both exclude / c 7 0 . Thus 7 0 is minimal. Hence it belongs
to Ilr.

No further / belongs to Ilr: Any other / of n r must be disjunct with
7 0 and with all (t), i in 7 — 7 0 , (use (43 :F) in 43.3.2.). As the sum of these
sets is 7, this would necessitate / = © — but © is not an element of II r
(cf. the beginning of 43.3.2).

Thus the proof is completed.

51.7.3. Combination of (43 :K) in 43.4.1. with (51 :N) gives: 1

(51:0) A simple game T is indecomposable if and only if 7 0 = 7, i.e.

if and only if all its participants are significant.

We conclude by proving:

(51 :P) A simple game T possesses precisely one /-constituent which

is simple and indecomposable: That with / = 7 0 .

Proof: The 7 0 -constituent can be formed and is simple by (51 :M).

Now consider a simple /-constituent. Then it has, by (51 :K), the same
W m and the same significant elements as T itself, — hence the latter form
the set 7 0 . So the indecomposability of the /-constituent is by (51:0)
equivalent to / = 7 0 .

We call the 7 0 -constituent A 0 of T its kernel. All other participants —
i.e. those of 7 — 7 0 — are “dummies.” (Cf. (51 :M) or (51:N), and the
last part of 43.4.2.). Hence all that matters in the game T takes place in
its kernel A 0 ; to see this, it suffices to apply the first remark in 51.6.

52. The Simple Games for Small n
52.1. Program: n — 1, 2 Play No Role. Disposal of n = 3

52.1. Our next objective is the enumeration of all simple games for the
smaller values of n. We propose to push this casuistic analysis so far as is
necessary to produce the examples referred to in 50.2. (cf. footnote 2 on

1 Or more directly of (43:K) in 43.4.1. with (51 :L), (51 :M).

468 SIMPLE GAMES

p. 434), 50.7.2., (cf. footnotes 2, 3, 4 on p. 443), 50.8.2. (cf. footnotes 1, 2, on
p. 445).

Since every simple game is essential, we need only consider games with
n £ 3.

For n = 3 the situation is this: The (unique) essential three-person game
is simple and it has the symbol [1,1,1]*. 1

So we can assume from now on that n ^ 4.

62.2. Procedure for n ^ 4 .‘^The Two-element Sets and Their Role in Classifying the W m

62 . 2 . 1 . Let an n ^ 4 be given. We wish to enumerate all simple games
with this n. In order to do this it is convenient to introduce a principle of
further classification of these games which is very effective for the smaller
values of n.

The enumeration in question amounts to the enumeration of the sets
W m for which we have various characterizations available — e.g. that one of
(51 :G) in 51.4.1.

Consider the smallest sets which may belong to W m . Since (51:G:d)
loc. cit. excludes the empty set and the one-element sets from W m } this
means considering the two-element sets in W m . These sets possess the
following property:

(52: A) A two-element set belongs to W m if and only if it belongs to

W . 2

Proof: The forward implication is obvious. Now assume conversely,
that the two-element set S belongs to W. The proper subsets of S are
empty or one-element sets, hence not in W . Therefore S belongs to W m .
We propose to classify according to two-element sets in W m .

62 . 2 . 2 . Conceivably W m may contain no two-element sets at all. We
denote this possibility by the symbol Co.

The next alternative is that W m contains precisely one two-element set.
By a permutation of the players 1, • * * , n we can make this set to be (1,2).
We denote this possibility by the symbol C i.

Further, W m may contain two or more two-element sets. Consider
two of these. By (51:G:a) loc. cit. they must have a common element.
By a permutation of the players 1, * • • , n we can make the common
element to be 1, and the two other elements of these two sets 2 and 3.

Se W m contains (1,2) and (1,3).

We denote the possibility that W m contains no further two-element sets
by the symbol C 2 .

62 . 2 . 3 . Now assume that W m does contain further two-element sets.
Assume furthermore that not all of them contain 1.

Consider therefore a two-element set of W m not containing 1. By
(51:G:a) loc. cit. it must have common elements with (1,2) and (1,3) — 1
being excluded, these must be 2 and 3 — so the set must be (2,3).

1 Cf. (50: A) in 50.1.1. and the last remark of 50.2.2.

1 1.e. a non-minimal set in W must have at least three elements.

SIMPLE GAMES FOR SMALL n 459

Thus (1,2), (1,3), (2,3) belong to W m . (To this extent we have perfect
symmetry in 1,2,3.)

Now consider any other two-element set which may belong to W m . It
cannot contain all three of 1,2,3; by a permutation of these players we can
arrange it so that the set in question fails to contain 1. Now it must have
common elements with (1,2) and (1,3) — 1 being excluded, these must be
2 and 3 — so the set must be (2,3), but we assumed it to be different from
(2,3) (among others).

Thus W m contains the two-element sets (1,2), (1,3), (2,3), and no others.
We denote this possibility by the symbol C*.

52.2.4. The remaining alternative is that W m contains other two-
element sets besides (1,2), (1,3), but that they all contain 1.

By a permutation of the players 4, • • • , n we can make these players to
be 4, • • • , k + 1 with a k = 3, • • • , n — 1.

Thus W m contains the two-element sets (1,2), (1,3), (1,4), • • • (l,i + 1),
and no others. We denote this possibility by the symbol C*.

52.2.5. It is convenient to bracket the cases C 0 , C i, C 2 of 52.2.2. and the
Ck f k = 3, • * • , n — 1 of 52.2.4. together: We then have the cases

Cky k = 0 , 1 , • • • , n - 1 .

In the case C* no wW m contains the two-element sets (1,2), • • • ,(l,fc + l),
and no others. By an additional permutation of the players 1, • • • , n l
we can replace these sets by (l,n), * * • , ( k,n ).

It is in this form that we are going to use the case C k) k = 0, 1 • • • , n — 1.
Now Ck contains the two-element sets (l,n), • • • , (A, n), and no others.

Besides these C* the only alternative is C* of 52.2.3. which we will not
transform.

62.3. Decomposability of the Cases C*, C n -t , C n -\

52 . 3 . 1 . Of all these alternatives three can be disposed of immediately:
C*, C n - 2 , On— i* We discuss these in a different order.

Ad 0*: Consider an S £ I. US contains two or more of 1,2,3 say (at
least) 1,2, then 5 2(1,2). (1,2) belongs to W ) hence S does too. If S

contains one or fewer of 1,2,3, say (at most) 1, then —(2,3). (2,3)

belongs to W, - (2,3) to L, hence S to L too. So we see: W consists precisely
of those S which contain two or more of 1,2,3. Hence W m consists precisely
of the sets (1,2), (1,3), (2, 3). 2 So (1,2,3) is the I 0 of 51.7. for this game.

In other words: The kernel of the game under consideration is a three-
person game with the participants 1,2,3, its W m consisting again of (1,2),
(1,3), (2,3). As mentioned before — for the last time in 52.1. — this game has
the symbol [1,1,1]*. The remaining n — 3 players, 4, • • • , n are
“dummies.”

iName *y C;U :: '.;«-i) ,cf - 28iL

* These were the two-element sets of W m by definition — but we have now shown that
they exhaust W m completely.

460

SIMPLE GAMES

So we see:

Case C* is represented by precisely one game: The three-person game
[1,1,1]*, with the necessary number (n — 3) of “dummies.”

52.3.2. Case C n -i: Consider an S £ I. Assume first, that n belongs
to £. If S has no further elements, then it is the one-element set (n), and
so in L. If S has further elements, say i = 1, • • • , n — 1, then S 2. (i, n) .
Now this (i, n) belongs to W ) hence S does too. In other words : if n is in S ,
then S belongs to W, except when S = (n). Applying this to — £ gives:
If n is not in £, then $ belongs to W ) when — S does not, i.e. if and only if
— S = (n), i.e. S = (1, • • • , n — 1).

Hence W consists precisely of these S : All sets containing n, except the
smallest one (n); no set not containing n, except the largest one,
(1,***, n — 1). One verifies easily that this W indeed fulfills the
requirements (49 :W*). Also that this game can be described as a weighted
majority game, all players 1, • • • , n — 1 having a common weight,
while player n has the n — 2 fold weight. I.e. this game has the symbol

U, • • • , 1, » - 2].

W m obtains immediately from W. It consists precisely of these S :
(1, n), • • • , (n — 1, n) and (1, • • • , n — l). 1 It is now easy to verify
that this game is homogeneous and normalized by a = 1. I.e. that a s = 1
(cf. 50.2.) for all the S of this W m . Hence we can write [1, * * • , 1, n — 2]*.

So we see:

Case C n - 1 is represented by precisely one game: The n- person game
[1, * • • , 1, n - 2]*.

52.3.3. Ad C n _ 2 *. Consider an S £ /. Assume first, that n belongs to S.
If S has no further elements other than possibly n — 1, then S £ (n — 1, n).
Now (n — 1, n) is not in W m , hence not in W (by (52 :A) in 52.2.1.). So
S is in L along with (n — 1, n). If S has further elements, other than n — 1,
say i = l, * • • , n — 2, then S 2 (t, n). Now this (i ) n) belongs to W
hence S does too. So we see: If n is in S , then S belongs to W, except when
S = (n) or (n — 1, n). Applying this to — S gives: If n is not in S then
S belongs to W when — S does not, i.e. if and only if —S = (n) or (n — 1, n),
i.e. S = (1, • • • , n — 1) or (1, • • • , n — 2).

Hence W consists precisely of these sets S : All sets containing n, except
(n), and (n — 1, n); no set not containing n, except (1, • • • , n — 1) and
(1, • • • , n — 2). One verifies easily that this indeed fulfills the require-
ment (49 :W*).

W m obtains immediately from W . It consists precisely of these S :
(1, n), •••,(»- 2, n), and (1, • • • , n - 2). 2 So (1, •••,»- 2, n)
is the Io of 51.7. for this game.

1 Thus the two-element sets in W m are (1, w), • • • , (n — 1, n), as it should be by
definition. The new fact is that the only further element of is (1, • • • , n — 1).

Note that this last set is not a two-element set only because of n ^ 4.

* Thus the two-element sets in W m are (1, »), • • • , (n — 2, n), as it should be by
definition. The new fact is that the only further element of W m is (1, • • • , n — 2).

For n — 4 this last set is also a two-element set, thereby falsifying the class of the
game. (It becomes C* instead of C„- j, i.e. C*.)

Hence this class (Cn- 1 ) is void, unless n £ 5.

SIMPLE GAMES FOR SMALL n

461

In other words: The kernel of the game under consideration is an
(n — l)-person game with the participants 1, • • • , n — 2, n, its W m
consisting again of (1, n), • • • , (n — 2, n), (1, - ■ • , n — 2). Thus
this is the case C n -2 for n — 1 players — the analogue of the case C n -i for n
players (replacing n by n — 1 !) discussed above. Hence it has the symbol
[1, • • • , 1, n — 3]*. The remaining player n — 1 is a “dummy.”

So we see:

Case C n -i is represented by precisely one game: 1 The (n — l)-person
game [1, • • • , 1, n — 3]* with one dummy.

52.4. The Simple Games Other than [1, • • • , 1, l — 2 U (with Dummies) : The Cases

C k y k « 0, 1, • • , n - 3

52.4. The results of 52.3. deserve to be considered somewhat further
and to be reformulated. We saw that for every l ^ 4 the homogeneous
weighted majority game of l players [1, • • • , 1, l — 2]* can be formed. 2 3
We can even form it for l = 3: Then it is the direct majority game of three
participants [1,1,1]*. So we will use it for all l ^ 3.

If n ^ 4 then we can obtain a simple n - person game by forming this
[1, * * * , 1, l — 2]* for any l = 3, • • • , n and adding to it the necessary
number of “dummies.”

The result of 52.3. was this: This game with l = 3, n, and (for n ^ 5)
n — 1 exhausts the cases C*, C n -i, C n - 2 .

The odd thing about this result is that these values of l do not exhaust
the full set of its possibilities l = 3, • • • , n (cf. above). That is to
say, they do this for n = 4, 5, but not for n ^ 6. There remain the
l = 4, • • • , n — 2 for n ^ 6. What is their significance?

The answer is this: Consider the game [1, • • • , 1, l — 2]* (l players)
with n — l “dummies.” Assume only l = 3, • • • , n and n ^ 4. The
W m consists of (1, l) y • • • , (l — 1, l) and (1, • • • , Z — 1).* Hence we
have case C* when / *= 3 and case Cz_ 1 when l = 4, • • • , n. 4 *

Thus we have in these games specimens from the cases C*, Cj, • • , C*_ 1 .
The result of 52.3. can now be formulated like this: The cases C*, C n -t, C„_i
are exhausted by the pertinent ones among these games. 6

We restate this conclusion:

(52:B) We wish to enumerate ail simple n- person games n ^ 4.

The game [1, • • • , 1, l — 2]* (/ players) with the necessary
number (n — l ) of “dummies” is a simple n- person game for
all l = 3, 4, • • • , n. Its case is C*, Cs, • • • , C n -\, respec-

1 For n ^ 5 — it is void for n * 4. Cf. footnote 2 on p. 460.

1 Cf. Case <?„_i above, with l in place of n.

3 We take players 1, • • • , l as the participants of the kernel [1, • • • , 1, l — 2]* and
players /-HI, * * * , n as “dummies.” This differs from the arrangement in case of
52.3. — where / — n — 1 and player w — 1 was “dummy” — by an interchange of players
n — 1 and n.

4 For / — 3, C* replaces C% since (1, • • • , Z — 1) is in this case a two-element set.

1 Hence C% is void for n ^ 4, since it occurs on the second list, but not on the first one.

Cf. 52.3.

462

SIMPLE GAMES

lively. All other simple n- person games (if any) are in the
cases Co, Ci, • • • , C n -i . 1

52.6. Disposal of n — 4, 5

62 . 6 . 1 . We will discuss the values n = 4, 5 fully and some characteristic
instances in n = 6, 7.

n = 4 is easily settled. By (52 :B) above, we need only investigate
Co, Ci for this n. In these cases W m contains ^ 1 two-element sets. How-
ever this is impossible: Since the complement of a two-element set is a two-
element set, there must be the same number of two-element sets in W and in
L. I.e. half of the total number, which is 6. So W contains 3 two-element
sets and the same is true for W m . 2

Thus the only simple games for n = 4 are those of (52 :B). We state
this as follows:

(52 :C) Disregarding games which obtain by adding dummies to

simple games of < four persons, 3 there exists precisely one simple
four-person game: [1,1, 1,2]*.

62 . 6 . 2 . Consider next n = 5. By (52 :B) above we must investigate
l = 0,1,2. In contrast to the n = 4 case, all of these represent concrete
possibilities.

C 0 : No two-element set is in W m i.e. in W. So they are all in L and
their complements, the three-element sets, are all in W. Thus W consists
of all sets of ^ three elements, and W m of all sets of three elements. Hence
this is the direct majority game [1,1, 1,1,1]*.

C\: (1,2) is the only two-element set in W m i.e. in W. Passing to the
complements: (3,4,5) is the only three-element set in L — i.e. the others are
in W. Thus W consists precisely of these sets: (1,2), all three-element sets
but (3,4,5), all four- and five-element sets. It is easy to verify that this
fulfills (49 :W*) and also that its W m consists of the following sets:

(1,2), (a,6,c), where a = 1,2, and b, c = any two of 3,4,5.

Now one shows without difficulty, that this game has the symbol

[ 2 , 2 , 1 , 1 , 1 ]*.

(1,2), (1,3) are the only two-element sets in W m , i.e. in W Passing
to the complements: (3,4,5), (2,4,5) are the only three-element sets ini, — i.e.
the others are in W. Thus W consists precisely of these sets: (1,2), (1,3),
all three-element sets but (2,4,5), (3,4,5), all four- and five-element sets.
It is easy to verify that this W fulfills (49 :W*) and also that its W m consists
of the following sets:

(1,2), (1,3), (2,3,4), (2,3,5), (1,4,5).

Now one shows without difficulty that this game has the symbol [3, 2, 2, 1,1]*.

1 All thpse cases which we succeeded in exhausting so far were void or contained
precisely one game. This is, however, not generally true. Cf. the first remark in 53.2.1.

* Owing to (52: A) in 52.2.1. This will be used in what follows continuously without
further reference*.

* I.e. to the unique simple three-person game [1,1,1]*.

NEW POSSIBILITIES OF SIMPLE GAMES 463

Hence the simple games for n = 5 are these three, and those of (62 :B).
We state as follows:

(62 :D) Disregarding games which obtain by adding dummies to sim-

ple games of < five persons, ‘there exist precisely four simple five-
person games: [1,1, 1,1,1]*, [1,1,1,2,21*,* [1,1,2, 2, 3]*,* [1,1,1, 1,3]*.

63. The New Possibilities of Simple Games for n ^ 6
53.1. The Regularities Observed for n < 6

53.1. Before we go further, let us draw some conclusions from the above
lists.

First: All simple games which we have obtained so far, possessed a
symbol, [vh, • • • , w n ]h, i.e. they were homogeneous weighted majority
games. Having verified this for n = 4, 5, the question arises whether it is
always true. As stated in footnote 3 on p. 443, this is not so; the first
counter-example comes for n = 6.

Second: So far every class C* which contained any game at all, contained
only one. This too fails from n = 6 on. (Cf. the first remark in 53.2.1.)

Third: One might think a priori , that there is great freedom in choosing
the weights for a homogeneous weighted majority game. Our lists show,
however, that the possibilities are very limited: One each for n = 3, 4, and
four for n = 5. 8 We emphasize that since our lists are exhaustive, this is
a rigorously established objective fact — and not a more or less arbitrary
peculiarity of our procedure.

Fourth: We can verify the statement of footnote 1 on p. 446 that while
the number of elements in W is determined by n (it is 2 n ~ 1 )> that one oi
W m may vary for simple games of the same n. This phenomenon begins
for n = 5.

For n = 3 : W has 4 elements, W m in the unique instance has 3. Foi
n = 4; W has 8 elements, W m in the unique instance 4. For n = 5: fFhas
16 elements, W m in the four instances 10,7,5,5, respectively.

Fifth: We can verify the statement of footnote 2 on p. 443, that the
equations (50:8) of 50.4.3., 50.6.2. (with U = W m ) may be more numerous
than their variables, and nevertheless possess a solution — i.e. a system ol
imputations in the ordinary sense. The former means that W m has > r
elements, the latter is certainly the case for homogeneous weighted majorit}
games ((50:K) in 50.8.1.).

We saw above that for n = 3, 4 W m necessarily has n elements, bul
for n = 5, it may have 10 or 7 elements as well. And all these games an
homogeneous weighted majority games. 4

»I.e. to [1,1, IK and to [1,1, 1,2]*.

* We permute the players of these games (belonging to O* and C 2 ) in order to have ai
increasing arrangement of weights.

3 Disregarding permutations of the players!

4 Thus we have the first counter-examples for n — 5: [1,1, 1,1,1]* (the direct majority
game) and [1,1, 1,2, 2]*.

464 SIMPLE GAMES

For a simple game, where these solutions do not exist, cf . the fifth remark
in 53.2.5.

53.2. The Six Main Counter-examples (for n - 6, 7)

53.2.1. We now pass to n = 6, 7. A complete exhaustion of these cases,
even of n = 6, would be rather voluminous. We forego it for this reason.
We will only give some characteristic instances of simple games in n = 6, 7
which illustrate certain phenomena which begin — as mentioned before — at
these n.

First: We mentioned in the second remark of 53.1. that for n = 6, a
case Ck may contain several games. Indeed, it is not difficult to verify that
the two homogeneous weighted majority games

[1,1,1,2,2,4]., [1,1,1,3,3,41.,

(cf. footnote 2 on p. 463) are different from each other and belong both to

Ct.

53.2.2. Second: We mentioned in the first remark of 53.1. that for n = 6
a simple game exists which is not a homogeneous weighted majority game,
i.e. one which does not possess any symbol [w i, • • • , w n ] h . By (50 :K) in

50.8.1. this is necessarily the case when there exists no main simple solution;
i.e. no system of imputations in the ordinary sense. (Cf. the fifth remark in

53.1. )

Such a game exists indeed, and it is even possible to differentiate further:
It is possible to find one which is nevertheless a weighted majority game
(without homogeneity!), i.e. which possesses a symbol [w if • • • , tt> n ], and
it is also possible to find one which does not even have that property.

We begin with the first mentioned alternative.

Put n = 6: Define W as the system of all those sets Ss / = (1, • • • ,6)
which either contain a majority of all players (i.e. have 4 elements), or
which contain exactly half of all players (i.e. have 3 elements), but a major-
ity of all the players 1,2,3 (i.e. ^ 2 of these). In other words: The players
1,2,3 form a privileged group as against the players 4,5,6 — but their privilege
is rather limited: Normally the overall majority wins; only in case of a tie
does the majority of the privileged group decide.

It is easy to verify that this W satisfies (49 :W*). The game is clearly a
weighted majority one: It suffices to give the members of the privileged
group (1,2,3) some excess weight over those of the others (4,5,6), which
must be insufficient to override an overall majority. Any symbol

[w, w, w, 1, 1, 1]

with 1 < w < 3 will do. 1

1 w > 1 is necessary, e.g. for 8 - (1,2,4) to defeat — S ■■ (3,5,6) (i.e. 2w + 1 > w + 2).
w < 3 is necessary, e.g. for S - (3, 4,5, 6) to defeat — S - (1,2) (i.e. w + 3 > 2 w).

NEW POSSIBILITIES OF SIMPLE GAMES

465

W m is quickly determined; it consists of these sets:

(1,2,3)

(a, 6, h) where a, 6 = any two of 1,2,3,

h = 4 or 5 or 6

(a,4,5,6) where a = 1 or 2 or 3. 1

The equations (50:8) of 50.4.3., 50.6.2. (with U = W m ) — which deter-
mine a main simple solution in the sense of 50.8.1. — are:

xi + x* + x$ = 6,

x a + Xb + Xh = 6, where a, 6 = any two of 1,2,3,

h = 4 or 5 or 6

x a + Xa + Xb + Xt = 6, where a = 1 or 2 or 3

These equations ( E \ ) cannot he solved. 2 Indeed, (E") with a = 1,6 = 2
and h = 4,5,6 shows that a* 4 = x b = x 6 ; (/?"') with a = 1,2,3 shows that

= x 2 = x 8 ; now (Z£J) gives 3xi = 6, Xi = 2; hence (/?") gives 4 + x A = 6,
Xi = 2; and then ( E "') gives 2 + 6 = 6 — a contradiction.

It should be noted that the ordinary economic aspect of this occurrence
would be this: (S") (i.e. (E")) shows that the services of players 4,5,6 can be
substituted for each other — hence they are of the same value. (Si")
(i.e. ( E [ ")) shows the same for 1,2,3. Now comparison of (S[) and (S")
shows that one player of the group 1,2,3 can be substituted for one player
of the group 4,5,6 — and comparison of (S") and (Si") shows that one player
of the former group can be substituted for two players of the latter. Hence
no substitution rate between these two groups can be defined at all. The
natural way out would be to declare some of the sets of W m enumerated in
(Si) to be “no profitable uses” of the players* services. In the sense of
50.4.3. this amounts to choosing U c W m (Cf. also 50.7.1. and footnote 4 on
p. 443). Whether in this game a6c W m can have the required properties
(cf. 50.7.1.) could be decided by a simple but somewhat lengthy combina-
torial discussion, which has not yet been carried out. The existence of
such a V is highly improbable, because it can be shown that it would have
mathematically unlikely characteristics if it existed.

This game is also very peculiar in another respect: It is possible to prove
that there exists no solution V which contains only a finite number of
imputations and which possesses the full symmetry of the game itself;
i.e. invariance under all permutations of the players 1,2,3 and under all
permutations of the players 4,5,6. We do not discuss this rather lengthy
proof at this place.* Thus the type of solution which one would term the
natural one does not exist.

1 Thus W m has 1 + 9 *f 3 - 13 elements.

* They are 13 equations in 6 variables, but this in itself is not necessarily an obstacle,
as the fifth remark in 53.1. shows.

* Whether any finite solution V exists at all, is not known. We suspect that even
this question will be anwered in the negative.

/ (E[):
(Ei) TO:

( m-.

466

SIMPLE GAMES

This is an indication of how extremely careful one must be in terming
extraordinary solutions “unnatural,” or in trying to exclude them.

63 . 2 . 3 . Third: Let us now consider the second alternative referred to
in the second remark above: A simple game for n = 6, which is no majority
game at all — i.e. which has no symbol [wi, • • • , w n \* This alternative
itself can be subdivided further: It is possible to find a game such that it
possesses a main simple solution (cf. above) — and it is also possible to find
one that has no main simple solution.

Consider the first case:

Putn = 6. Define W as the system of all those sets S(£/ = (1, • • • ,6))
which either contain a majority of all players (i.e. have ^ 4 elements), or
which contain exactly half (i.e. have 3 elements), but an even number of the
players 1,2,3 (i.e. have 0 or 2 of these). Comparing this with the example
in the second remark above, this observation must be made: The players
1,2,3 still form a group of special significance, but it would be misleading to
call their significance a privilege — since their absence from the tying (i.e.
three-element) set S is just as advantageous as their strong representation
(presence of precisely two of them), and the presence of all of them just as
disastrous as their weak representation (presence of precisely one of them).
They bring about a decision not by their presence in S but by an arithmetical
relation: 1

It is easy to verify that this W fulfills (49 :W*) in 49.6.2. 2 3

Let us now determine W m . Since W contains all ^ four-element sets,
no ^ five-element set can be in W m . Consider now a four-element set in W.

If the number of players 1,2,3 in it is even, remove from it a player 4 or 5
or 6.* If the number of players 1,2,3 in it is odd, remove from it a player 1
or 2 or 3. 4 * At any rate a three-element subset with an even number of
players 1,2,3 obtains — i.e. one in W. So no four-element set can be in W m .
Hence W m consists of the three-element sets in W. These are:

(St)

(Si):

(«'):

(4,5,6)
(a, 5, h)

where a, 6 = any two of 1,2,3;
ft = 4 or 5 or 6. 6

1 Note also that the group 4,5,6 has a similar significance: Since S must have three
elements (in order that these criteria become operative), the statement that an even
number of 1,2,3 is in S is equivalent to the statement that an odd number of 4,5,6 is in S.

This lends further emphasis— if any be needed — to our frequently made observation
concerning the great complexity of the possible forms of social organization, and the
extreme wealth of attendant phenomena.

* Note in particular that always one of S and — S belongs to W : This is evident if one
of the two has ^ 4 elements (and so the other 2). Otherwise both S and —S have

3 elements. Hence one of them contains an even number of players 1,2,3 and the other
an odd one.

1 This is possible, as 1,2,3 are only 3 players.

4 This is possible, as 4,5,6 are only 3 players.

1 Thus W m has 1 -f 9 — 10 elements.

NEW POSSIBILITIES OF SIMPLE GAMES

467

If this game had a symbol [w 1( • • • , w n ], then there would be
Wi > 5) w * f° r all <S in W.

i in 8 t not in 8

Apply this to the sets of W m enumerated in (St). This gives in particular:

W 4 + Ws 4" > 101 -f- Wt -f* ttfj,

Wi + Wi + Wt > Wz + Wi + Wi,

Wi + Wi -f Wt> > Wi + Wi + Wa,

Wi + Wi + Wi > Wi + Wi + Wt.

Adding these inequalities gives:

2 (wi + Wi + Wt + W 4 + wi, + Wi) > 2 (wi + Wi + w* + W 4 + w* + w t ),

a contradiction.

The equations (50:8) of 50.4.3., 50.6.2. (with U = W m ) — which deter-
mine a main simple solution — on the other hand are:

( E'z )• Xi + Xf, + xt = 6,

(Ei) < {E'i): x a + x b + x h = 6, where a, b = any two of 1,2,3;

h = 4 or 5 or 6.

They are obviously solved by Xi = • • • = = 2. 1

In the ordinary economic terminology one would have to say that the
structural difference between the groups of players 1,2,3 and 4,5,6 cannot be
expressed by weights and majorities, and that as far as values are concerned,
there is no difference.

63.2.4. Fourth: Note that the above example is also suited to establish
the difference between the homogeneous weighted majority principle and
the existence of a main simple solution, as discussed in 50.8.2. Indeed,
it is an instance of = in (50:21) loc. cit. : Since Xi = • • • = x* = 2 (cf.
above), so

n

5) Xi = 12 = 2 n.

»- 1

63.2.6. Fifth: Now consider the second case described in the third
remark above: A simple game for n = 6, for which neither a symbol

[Ml, * * • , Wn]

nor a main simple solution exists.

Compared with the two previous examples — given in the second and
third remark above — this one is based on less transparent principles. This
is not surprising since now all our simplifying criteria are to be unfulfilled.
This is the example:

Put n = 6. Define W as the system of all those sets S(£ I = (1, • * * ,6))
which contain either a majority of all players (i.e. have ^ 4 elements), or

1 It is easily seen that this is their only solution.

468

SIMPLE GAMES

which contain exactly half (i.e. have 3) elements, and fulfill the following
further condition: Either S contains player 1 , but it is not (1,3,4) or (1,6,6) 1 —
or S is (2,3,4) or (2,5,6). 2 * s

It is easy to verify that this W satisfies (49:W*) in 49.6.2.

W m can be determined without serious difficulties. It turns out to
consist of these sets:

(SI):

( 1 , 2 , 6 )

where b = 3 or 4 or

5 or 6

(-Si'):

(l,o, 6 )

where a = 3 or 4, b

= 5 or 6 4

OS

( 2 ,p,g)

where p = 3, q = 4,

or p = 5, q = 6. 4

(S« y ) :

(3, 4,5,6)’

If this game had a symbol [w h • • • , w n ] } then there would be
^ w i > Wi for all S in W.

i in S i in — S

Apply this to the sets of W m , enumerated in ( S 3 ). This gives in particular:

Wi + W 3 + W b > W 2 + Wa + w 6y

Wi + Wa + We > w 2 + w 3 + w b ,

W 2 + W 3 + Wa > Wi + Wh + Way

w 2 + w b + w 6 > Wi + w 3 + w A .

Adding these four inequalities gives :

2 (wi + w 2 + w 3 + w A + w b + We) > 2(wi + W 2 + w 3 + Wa + w b + We),
a contradiction.

The equations (50:8) of 50.4.3., 50.0.2. (with U = W m ) — which deter-
mine a main simple solution — on the other hand are:

f ( E 3 ) : x\ + x 2 + x b = 6, where 6 = 3 or 4 or 5 or 6,

(E 3 ) : Xi + x a + Xb = 6, where a = 3 or 4, b = 5 or 6,

( E z ) ( E 3 ')\ x 2 + x p + x q = 6, where p = 3, q = 4, or p = 5,

q ~ 6,

k (^s v ) : x 3 + x 4 + x 5 + X 6 = 6.

These equations (^ 3 ) cannot be solved . 8 Indeed (!?£') shows that x 3 = x 4
and Xg = x«, hence (£ 3 ") gives X 2 + 2x 3 = 6 , x 2 + 2x 6 = 6 , therefore

1 I.e. it is (1 ,a,b) with 0 = 2, b * 3 or 4 or 5 or 6; or with a — 3 or 4, b — 5 or 6.

* The complements of the previously excluded sets (1,5,6) and (1,3,4).

* If this last exception — concerning (1,3,4), (1,5,6) and (2,3,4), (2,5,6) — were omitted,
then W would be defined by this principle: The player 1 is privileged — normally the over-
all majority wins, but ties are decided by player 1.

It is easy to verify that this is simply the game [2, 1,1, 1,1,1]*. I.e. this case is even
simpler, than our — in some ways, analogous — example in the second remark above —
since the privilege existing here has a numerical value in the conventional sense.

Thus the complicating exception — concerning (1,3,4), (1,5,6) and (2,3,4), (2,5,6) — is
decisive in bringing forth the real character of our example.

4 Note that 0,6 vary independently of each other, while p,q do not!

• Thus W m has 4-h4+2-fl » 11 elements.

• They are 10 equations in 6 variables, cf. footnote 2 on p. 465.

NEW POSSIBILITIES OF SIMPLE GAMES

460

xz = x B , and so x% = x 4 = x 6 = x 6 . Now (Z7£ v ) gives 4x 3 = 6, x 3 = f
whence (E' s '), (#',") yield Xi + 3 = 6, x 2 + 3 = 6, i.e. Xi = x* = 3. Finally
(Z?J) becomes 3 + 3 + i = 6, — a contradiction.

As to the interpretation of this insolubility, essentially the same com-
ments are in order as at the corresponding point of the second remark above.

63 . 2 . 6 . Sixth: We have already referred to the difference between the
homogeneous weighted majority principle, and the existence of a main
simple solution, as discussed in 50.8.2. This was done in the fourth remark
above, where an example for = in (50:21) loc. cit. was given. We will now
give an example for > in (50:21) loc. cit.

Since we found that for n g 5 all simple games were homogeneous
weighted majority games, we must now assume n <z 6. We do not know
whether an example of the desired kind exists for n = 6 — the one which
will be given has n = 7.

Putn = 7. Define Was the system of all those setsS(£/ = (1, • • • ,7))
which contain any one of the 7 following three-element sets: 1

(Sa): (1,2,4), (2,3,5), (3,4,6), (4,5,7), (5,6,1), (6,7,2), (7,1,3)

The principle embodied in this definition can be illustrated in various
ways.

This is one: The 7 sets of (S 4 ) obtain from the first one —(1,2,4) — by
cyclic permutation. I.e. by increasing all its elements by any one of the
numbers 0,1, 2, 3, 4, 5, 6 — but all three by the same one — provided that the
numbers 8,9,10,11,12,13 are identified with 1,2, 3, 4, 5, 6 respectively. 2

In other words: They obtain from the set marked x x x on Figure 89,
by any one of the 7 rotations which this figure allows.

Another illustration: Figure 90 shows the players 1, • • • , 7 in an
arrangement in which it is feasible to mark 7 sets of (S 4 ) directly. They are
indicated by the 6 straight lines and the circle O. 1

The verification, that this W fulfills (49 :W*) is not difficult, but we
prefer to leave it to the reader if he is interested in this type of combinatorics.
W m consists obviously of the 7 sets of (S 4 ).

It is easy to show — along the lines given in the third and fifth remarks
above — that this is not a weighted majority game. We omit this discussion.

The equations (50:8) of 50.4.3., 50.6.2. (with U = W m ) — which deter-
mine a main simple solution — on the other hand are:

(Ea): x a + x b + x c = 7, where (a,6,c) runs over the 7 sets of (<S 4 ).

They are obviously solved by xi = • • * = X7 = i. 4

1 Thus W m has 7 elements.

* In the terminology of number theory: Reduced modulo 7.

* The reader who is familiar with projective geometry will note that Figure 90 is the
picture of the so-called 7 point plane geometry. The seven sets in question are its
straight lines, each one containing 3 points, and the circle O also rating as such.

One should add that other projective geometries do not seem to be suited for our
present purpose.

4 It is easily seen that this is their only solution.

470 SIMPLE GAMES

Now we can establish that > holds in (50:21) in 50.8.2. Indeed:

2) Xi = V- > 14 = 2n.

»- i

As the games discussed in the second, third and fifth remarks, this one
too corresponds to an organizational principle that deserves closer study.
In this game the sets of W m , i.e. the decisive winning coalitions are always
minorities (three-element sets). Nevertheless, no player has any advantage
over any other: Figure 89 and its discussion show that any cyclic permuta-
tion of the players 1, • • * , 7 — i.e. any rotation of the circle of Figure 89 —
leaves the structure of the game unaffected. Any player can be carried in
this manner into any other player's place. 1 Thus the structure of the game

is determined not by the individual properties of the players 2 — all are, as
we saw, in exactly the same position — but by the relation among the players.
It is, indeed, the understanding reached among 3 players who are correlated
by 0S 4 )* which decides about victory or defeat.

64. Determination of All Solutions in Suitable Games

64.1. Reasons to Consider Other Solutions than the Main Solution in Simple Games

64.1.1. Our discussion of simple games thus far placed most emphasis
upon the special kind of solutions discussed in 50.5.1.-50.7.2. and particu-
larly on the main simple solution of 50.8.1. On the basis of what we have
learned in the previous sections — especially from the examples of 53.2. — this
approach does not appear to do justice to all aspects of our problem.

1 The game is nevertheless not fair in the sense of 28.2.1., since e.g. the two three-
element sets (1,2,4) and (1,3,4) act differently: The former belongs to W , the latter to L.
(So in the reduced form of the game, with y - 1, the v(S) of the former is 4, and that of
the latter is —3.)

* Which the rules of the game might give them.

’There exist in this game no significant relations between any two players: It is
possible to carry any two given players into any two given ones by a suitable permutation
(of all players 1, * • • , 7) which leaves the game invariant.

ALL SOLUTIONS IN SUITABLE GAMES

471

To begin with, we have seen that we cannot expect all simple games to
have solutions of the type mentioned. Already for n » 6 a wealth of new
possibilities emerged. This is significant, since 6 is a sizeable number
from the point of view of combinatorics, but a small one when viewed in
the context of social organization.

But further, even when these solutions exist, indeed even for the homog-
eneous weighted majority games, they do not tell the whole story. For the
most primitive specimen of that class, the essential three-person game
which as we know has the symbol [1,1,1]*, there exist many solutions.
And our discussion in 33. showed that they are all essential for our under-
standing of the characteristics and the implications of our theory — actually
some fundamental interpretations were first obtained at that point.

64 . 1 . 2 . Consequently it is important to determine all solutions of a
simple game and, as long as we are not able to do this for all simple games,
to do it for as many simple games as possible. In particular this should
be done for at least one simple game at each value of n. Such results would
provide some information about the structural possibilities and principles
of classification of solutions for ^participants.

It is true that this information would be equally welcome if it could be
obtained for other than simple games. However the simple games possess
a manifest advantage over all others when solutions are to be determined
systematically: For simple games the so-called preliminary conditions of

30.1.1. cause no difficulties (cf. 31.1.2.), since there every set S is certainly
necessary or certainly unnecessary (cf. 49.7.).

It is equally true that the determinations which we envisage would only
provide information concerning a few isolated cases. But they would
nevertheless cover all n — i.e. enable us to vary n at will. This is bound to
lead to essential insights.

64.2. Enumeration of Those Games for Which All Solutions Are Known

64 . 2 . 1 . Let us take inventory of the cases for which we already know
all solutions of a game. There are three:

(a) All inessential games (cf. (31 :P) in 31.2.3., complemented by (31:1)
in 31.2.1.).

(b) The essential three-person game both in the old theory (excess
zero) and in the new one (general excess). (Cf. 32.2.3. for the former
and the analysis of 47.2.1.-47.7. for the latter.)

(c) All decomposable games — provided that all solutions of the con-
stituents are known. (Cf. (46:1) in 46.6.)

Clearly we can use the device (c) to combine the games provided by (a)
and (b) — thus obtaining games for which all solutions are known. 1 In
this process of building up (a) furnishes only “dummies” (cf. the end of

43.4.2. ), hence we may well dispense with it, since we want structural

1 This can also be expressed in the following way:

A given game r is the composite of its indecomposable constituents, according to
the definition of the decomposition partition at the end of 43.3. and (43 :E) eod. We

472

SIMPLE GAMES

information. Thus we are left with those games which are obtained by
iterated application of (c) to (b). In this way we can obtain games which
are the composite of essential three-person games. 1

64 . 2 . 2 . This gives n = 3fc~person games for which we know all solutions.
Since k is arbitrary, we can make n arbitrarily great. To this extent things
are satisfactory. However the fact remains that such an n- person game
is just a polymer of the essential three-person game — the players form in
reality sets of 3 which the rules of the game fail to link to each other. It
is true that our results concerning the solutions of decomposable games
show that a linkage of these sets of players is nevertheless provided for in
the typical solution — i.e. by the typical standard of behavior. But natu-
rally we want to see how the ordinary type of linkage, explicitly set by the
rules of the game, affects the organization of the players — i.e. the solutions
or standards. And we want this for great numbers of players.

Consequently we must look for further n- person games for which it is
possible to determine all solutions.

64.3. Reasons to Consider the Simple Game [1, • • • , 1, n — 2]*

64 . 3 . 1 . As pointed out above, we are going to look for these specimens
among the simple games. 2 Now it turns out that there is a certain simple
game for every n ^ 3, for which this determination can be carried out.
This is the only n- person game, of a general n, for which we succeeded thus
far in the general determination. This obviously gives it a position of
special interest. We will also see that it permits interesting interpretations
in several respects.

The game in question has already occurred in 52.3. and in (52 :B) of
52.4. It is the homogeneous weighted majority game [1, • • • , 1, n — 2]*,
(n players).

know from (43 :L) in 43.4.2. that the sets of participants into which this partition sub-
divides them, are sets of 1 or ^ 3 elements.

The simplest possibility is therefore that they are all one-element sets. According
to (43 :J) in 43.4.1. this means that the game is inessential — i.e. it takes us back to the
case (a) above.

The next simplest possibility is that they are all one- or three-element sets. These
are exactly the games which we can form according to (c) from (a) and (b). I.e. it is
for these that we know all solutions.

This is satisfactory since it shows that a classification based on the sizes of the
indecomposable constituents (i.e. of the elements of the decomposition partition, cf.
(43 L) in 43.4.2.) is a natural one : Our progress in obtaining all solutions follows precisely
the lines drawn by it.

It also stresses how limited these results are : It is indeed a very special occurrence
when a game is decomposable at all. (Remember the defining equations of (41:6) or
(41:7) in 41.3.2., according to the criterion at the end of 42.5.2.!) The typical n-person
game is indecomposable and cannot be reached by means of (c).

1 By application of strategic equivalence we can assume them all to be in the reduced
form. But denoting their y by 71 , • • • , y k respectively, we cannot expect to make
them all equal to 1 by a change of unit (unless A; — 1 ). Indeed, their ratios 71 : • • • : y k
are unaffected by changes of unit.

* For this reason we use the old theory, i.e. excess zero. Cf. the third and fourth
remarks in 51.6.

473

THE SIMPLE GAME [1, * • • , 1, n - 2]*

54 . 3 . 2 . As discussed in 52.3. in this game the minimal winning coalitions
are these S: (1, n), • - • , (n — 1, n) and (1, • • • , n — 1). I.e. player n
wins as soon as he finds any ally at all, but if he remains completely isolated
then he loses. 1 This result invites some remarks:

First: The statement of this rule suggests strongly that player n is in a
privileged position: He needs only one ally to win, while the others need
each other without exception. Actually the situation is this: Player n
needs a coalition of two, the others together need one of n — 1, hence a
privilege exists only if n — 1 > 2, i.e. n ^ 4.

For n = 3 there is, indeed, no difference between the three players:
We have then the game [1,1,1]*, the unique essential three-person game
which is obviously symmetric.

Second: The privilege of player n is as extensive as a privilege can be:
We required that n must find at least one ally in order to win and it would
not have been possible to require less. 2 It is impossible to specify that n
can win without an ally, i.e. to declare that the one-element set in) to be
winning — this is incompatible with the essentiality of the game. (This was
discussed extensively in 49.2.)

65. The Simple Game [1, • • • , 1, n — 2]*

66.1. Preliminary Remarks

66 . 1 . The determination of all solutions of the game which we discussed
above will show that they fall into a complex array of classes, exhibiting
widely varying characteristics. These create an opportunity for the
interpretations we have alluded to previously. We will discuss some of
them, while further discussions along the same line will probably follow in
subsequent investigations.

The exact derivation of this complete list of solutions will be given in the
sections which follow (55.2.-55.11.). This derivation is of not inconsider-
able complexity. We are giving it in full for the same reasons as the
analogous one concerning the solutions of decomposable games in Chapter
IX. : The proof itself is a convenient and natural vehicle for certain interpre-
tations. It presents at several stages an opportunity to bring out verbally
the emerging structural features of the organizations under consideration.
In fact this circumstance will be even more pronounced in the proofs of this
chapter than in those of Chapter IX.

66.2. Domination. The Chief Player. Cases (I) and (II)

55 . 2 . 1 . After these preliminaries we proceed to the systematic investiga-
tion of the game [1, • * • , 1, n — 2]* (n players). Assume that it is in the
reduced form, normalized by y = I.

1 As every one-element set must.

1 We stated above that player n is not at all privileged in this game when n « 3 —
and now we state that he is as privileged as he possibly can be! Yet n — 3 is no excep-
tion from this statement: Since there exists only one essential three-person game, the

474

SIMPLE GAMES

We begin with an immediate observation on domination:

(55: A)

For a = {ai,

> «»}, P = [Ph

y Pn

a H j8

if and only if either

(55:1) a n > Pn and > ft for some i = 1, • • * , n — 1,

or

(55:2) ai > & for all i = 1, • • • , n — 1.

Proof: This coincides with (49 :J) in 49.7.2., since W m consists of the sets
(1, n), • • • , (n - 1, n) and (1, • • • , n - 1).

n n

Note that £ = £ 0* = 0 permit us to infer from (55:2) the validity

»- i i

of

(55:3)

Hence :

> C_j — ►

a n ^ Pn>

(55 :B)

« % fi

necessitate

ctn 7* Pn -

Proof: By symmetry we need only consider a H 0 . We saw that this
implies (55:1) or (55:3), hence at any rate a n j* f $ n .

These two results, simple as they are, deserve some interpretative
comment.

We discussed in 54.3. that the player n has a privileged role in this
game. 1 He is in a situation which is comparable to that of a monopolist ,
with the inescapable limitation (cf. the second remark loc. cit.) that he
must find at least one ally. I.e. a general coalition of all others against
him — but nothing less than that — can defeat him. We will call him the
chief player in this game. 2

55.2.2. These circumstances are brought out clearly in (55:1) and (55:2).
One may say that (55:1) is the direct form of domination by the chief player
and an arbitrary ally (any player i = 1, • • * , n — 1) while (55:2) may be
termed a state of general cooperation against him. (55:1), (55:3) or (55:B)
show that in a domination the chief player is certainly affected: Advan-
tageously in (55:1) (the direct form of domination with the chief player),
adversely in case (55:2) (the general cooperation against the chief player).
Any other player can be unaffected, left aside, in a domination.*

position in which a player finds himself there may as well be called the best possible one —
since it is the only one there is.

1 Except the case n — 3, about which more will be said later.

* As to the case n — 3, the end of the first remark in 54.3. should be kept in mind.

1 I.e. It may happen for an • - 1, • • • , n — 1 that a H 0 and a< - 0*. This is

actually only possible when n £ 4, cf . again the observations concerning » ■■ 3.

THE SIMPLE GAME [1, • • • , 1 , n- 2] h
55.2.3. Now consider a solution V of this game. 1 * Form

475

Clearly

Max-* . a n — u,

<» in V

Min-* . a„ =
mV

— 1 £ q ^ w.

The meaning of a> is plain: They represent the. worst and the best
possible outcome for the chief player, within the solution V.

We distinguish two possibilities:

(I) tf = W,

(II) U < 01.

65.3. Disposal of Case (I)

• 66.3.1. Consider the case (I). This means that for all a in V
(55:4) a n = d>,

i.e. that the chief player obtains the same amount under all conditions
within the solution. In other words: (I) expresses that the chief player
is segregated in the game in the sense of 33.1. Considering the central
role of the chief player it is not unreasonable that the first alternative
distinction in our discussion should proceed along this line. 3

66.3.2. Let us now discuss V in case (I).

— ►

(55 :C) V is precisely the set of all a fulfilling (55:4).

1 In the sense of the old theory, cf. footnote 2 on p. 472.

1 That these quantities can be formed, i.e. that the maximum and minimum exist and
are assumed can be ascertained in the same way as in footnote 1 on p. 384. Cf. in partic-
ular (*) loc. cit.

* The reference to 33.1. re-emphasizes that this procedure is analogous to that one of
the essential three-person game.

This will appear even more natural if it is recalled that the essential three-person
game is a special case of the one we consider now — pertaining to n — 3. (Cf. e.g. the
end of the first remark in 54.3.)

Closer consideration of the case n =* 3 shows, however, that this analogy suffers
from a rather unsatisfactory limitation : In this case the game is really symmetric, and so
any one of the three players could have been called the chief player. (Cf . also footnote 2
on p. 474.) In 33.1. the segregation in question was indeed applicable to any one of the
three players, and now we have arbitrarily restricted it to player n!

Yet there is no way so far to apply this to the other players too if we want our
discussion to cover all n ^ 3 (and not only n — 3) : For n 4 the chief player and his
role are unique.

The only sense in which this situation can be accepted — temporarily — is that of
keeping in mind that case (II) must in fine turn out to be a composite one.

Thus for n « 3 comparison with the classification of 32.2.3. — which is analysed in
33.1. — shows this: Our case (I) is one of the possibilities of (32: A) there: discrimination
against player 3. Our case (II), on the other hand, covers the other two possibilities of
(32 :A): discrimination against players 1,2 — together with (32 :B), the non-discriminatory
solution. So (II) is really an aggregate of 3 possibilities when n » 3.

This scheme will, indeed, generalize for all n. Cf. (e) in the fourth remark of
55.12.5.

476

SIMPLE GAMES

Proof : We know already that all a of V fulfill (55:4). If, conversely, a
p fulfills (55:4), then every a of V has a n = Pn, hence (55 :B) excludes

a H P . Hence p belongs to V.

Thus V is determined easily enough, but we must now answer the
converse question: Given an w | — 1 , is the V defined by (55:4) (i.e. by
( 55 :C)) a solution? I.e. does it fulfill (30:5:a), (30:5:b) of 30.1:1.?

Now (55 :B) and (55:4) exclude a h P for a , P in V, hence (30:5:a)
is automatically satisfied. Therefore we need only investigate (30:5:b) of
30.1.1. I.e. we must secure this property:

( 55 : 5 ) If p n t* w, then a H p for some a with a n = w.

More explicitly: We must determine what limitations (55:5) imposes
upon o).

The p n 9* of (55:5) can be classified:

(55:6) p n > u,

(55:7) p n < «.

We show first:

(55 :D) In the case (55:6) condition (55:5) is automatically fulfilled.

Proof: Assume p n > w, i.e. p n = & + « > 0. Define

— >

a = |«i, ■ ■ • , a,|

by on = fa -\ 7 for i = 1 , • • • , n — 1 , and a n = p n — t = o>. a is

— ► — >

an imputation of the desired kind with a H P by (55:2).

Thus only the case (55:7) remains. Concerning this case we have:

(55 :E) For w = —1, (55:7) is impossible.

Proof: Pn £= —1, hence not p n < w = -1.

The possibility w > — 1 is somewhat deeper . 1

(55 :F) Assume o> > — 1 and case (55:7). Then condition (55:5)

is equivalent to w < n — 2 — — -L -*

1 d> — — 1 means that the chief player is not only segregated but also discriminated
against (by V) in the worst possible way. (Cf. 33.1.)

Thus « • — 1 gives a solution outright, while u> > — 1 necessitates the more
detailed analysis of (55 :F). This is not surprising: An extreme form of discrimination
is a more elementary proposition and requires less delicate adjustments than an inter-
mediate one.

THE SIMPLE GAME [1, • • • , 1, n - 2]*

477

Proof: Assume p n < w. For an a with a n = w, (55:3) of 55.2.1. is

excluded, i.e. domination a h p must operate through (55:1) (and not
(55:2)!) in (55 :A). Since a n > p n , this condition amounts merely to

(55:8) a< > ^ for some i = 1, • • • , n — 1.

I Thus (55:5) requires the existence of an imputation a with a n = u>
and (55:8).

Consider first (55:8) for a fixed i = 1, • • • , n — 1. Then this con-
dition and a n = o> can be met by an imputation a if and only if Pi and «
add up with n — 2 addends — 1 to <0. I.e. Pi + a> — (n — 2) < 0,
Pi < n — 2 — co. Consequently (55:8) is unfulfillable for all i = 1, • • • ,
n_r- 1, if and only if

(55:9) Pi ^ n — 2 — for all i = 1, • • • , n — 1.

(55:5) expresses that this should happen for no P with p n < o>. I.e.

no imputation p could have (55:9) together with — 1 ^ p n < w. 1 This
means that n — 1 addends n — 2 — w and one addend — 1 must add up

to > 0. I.e. (n — 1 )(n — 2 — w) — 1 > 0, n — 2 — w> — p and so

ft — i

« < ft — 2 as desired.

ft — 1

Combining (55 :E), (55 :F) and recalling (55 :D) and the statements
made concerning (55:5) and (55:6), (55:7), we can summarize as follows:

(55 :G) Let o> be any number with

-1 ^ < n - 2

ft ~ 1

Form the set V of all a with

CLn = W . 2

These are precisely all solutions V in the case (I).

1 We are assuming that (55:9) implies £» ^ — 1 for i = 1 , • • • , n — 1 . This means
n — 2 — w ^ —1, w ^ n - 1. Indeed « > n — 1 must be excluded since it makes
(55:4) unfulfillable by imputations: u and n — 1 addends —1 would then add up to
> 0 .

Therefore the hypotheses of (55 :F) imply this: £ £ n — 1.

* In pursuance with the parallelism with the discussion of the special case n ■■ 3 in
33.1. — referred to in footnote 3 on p. 475 — we note that this corresponds to the cloc. cit.

For ft — 3 our ft — 2 — - — — becomes the i occurring there.

478

SIMPLE GAMES

The first values of the quantity n — 2 —

1

7 are:

n — 1

n

3

4

5

6

n — 2 — \

n — 1

|-0.5

| - 1.67

T ~ 2 75

4

Figure 91.

66.3.3. The interpretation of this result is not difficult:

This standard of behavior (solution) is based on the exclusion of the
chief player from the game. This makes the distribution between the
other players quite indefinite — i.e. any imputation which gives the chief
player the “ assigned” amount u> belongs to the solution. The upper limit

of the “assigned” amount d>, n — 2 — — could also be motivated

following the lines of 33.1.2., but we will not consider this question.

55.4. Case (II) : Determination of V

66.4.1. We now pass to the considerably more difficult case (II). (Cf.
the last part of footnote 3 on p. 475.) We then have

— 1 ^ y < a?.

This suggests the following decomposition of V into three pairwise
disjunct sets:

— ►

V, set of all a in V with a n = a>,

V, set of all a in V with a n = d>,

V*, set of all a in V with a> < a n < a>.

By the very nature of w, w (cf. the beginning of 55.2.3) V, V cannot be
empty — while we cannot make such an assertion concerning V *. 1
66.4.2. We begin by investigating V.

(55 :H) If a belongs to V and )3 to Vu V*, then ^ ft for all

i = 1, • • • , n — 1.

Proof: Otherwise ft > a* for a suitable i = — 1. Now

a n = w, ft* > y, so ft > ot n ; hence 0 H a by (55:1), which is impossible.
— ► — ►

since a , f) belong to V-

1 V* is actually empty in the case considered preceding (55 :V).

Form

THE SIMPLE GAME [1, * - • , 1, n — 2] h

479

cti = Min-> . a, for i = 1 • • • , n — l. 1
a mV

Now (55 :H) gives immediately:

(55:1) If 0 belongs to Vu V*, then

gfi ^ & for all i = 1, • • • n — l. 2

We prove further

n — 1

(55 : J) X ® ^ 0.*

t-1

n — 1

Proof: Assume that ^ g; -f a? < 0. Then we can choose 7< > a* for

t = 1, • • * , n — 1, 7n = a> with ^ 7, = 0, forming the imputation

t- i

— ►

7 = {71, * * • , 7n } • 4

V is not empty, choose a 0 from V. Then by (55:1) ft ^ < 7< for
all i = 1, • • • , n — 1, hence by (55:2) 7 H j8. As 0 belongs to V,
this excludes 7 from V.

Hence there exists an

a in V with a £-• 7 .

If a belongs to V f

then oc n = = 7 n , hence a h 7 contradicts (55 :B). So a must belong
to S? u V*. Now by (55:1) a* ^ a, < 7 , for all t = 1 , • * * , n — 1 .

But both (55:1) and (55 :2) in (55: A) require — since a H 7 — that a* > 7,
for at least one i = 1, • • • , n — 1. Thus we have a contradiction.

Now the determination of V can be completed:

(55 :K) V has precisely one element:

a 0 = {ai, * • * , a n -ij w).

1 That these quantities can be formed, i.e. that these minima exist and are assumed
can be ascertained in the same way as in footnote 1 on p. 384. Cf. in particular (*)

loc. cit. What is stated there concerning V, is equally true for V, J;he intersection of V

with the closed set of the a with a* — tf .

* Note that this cannot be asserted for the 0 of V since 0 * may exceed the minimum
value Cf., however, (55 :L).

* Cf. however, (55:12) below.

4 Note that by their definitions all g» £ —1, (*' — 1, • • • , n — 1) and « ^ — 1,
hence all our 7 < £ — 1 , (t - 1 , • • • , n — 1 , n).

480

SIMPLE GAMES

Proof: Let a = («i, • • • , a„_i, a„) be an element of V. Then

(55:10)

I «, £ «,

( = W,

for i = 1, • • • , n — 1,

by the definition of these quantities. Now ^ = 0 and by (55 :J),

»-i

n — 1

£ cti + o) 0, hence > is excluded from all inequalities of (55:10).

i-1 ~

Thus

(55:11)

i.e.

ct{ = cti for i = 1, • • • , n r 1,
«» = co,

{«1| • * * i «n-l, «n) = {«!, • • • , «n-l, «)•

So V can have no element other than {ai, • • * , a n _i, <o). Since V

is not empty, this is its unique element.

— >

65.4.3. Note that as a 0 = {ai, • • • , a„_i, «} belongs to V, it is
necessarily an imputation. So we can strengthen (55 :J) to

(55:12)

n — 1

5) a< + w = 0.

We can also strengthen (55:1):

(55 :L) If p belongs to V, then

a» ^ P% for all i = 1, ••*,»— 1.

Proof: For p in V u V* this has been stated in (55:1), for p in V
(55 :K) yields even /?< = a,-.

We conclude this part of the analysis by proving:

(55 :M) a = -1.

Proof: Assume « > — 1, i.e. « = — 1 + «, « > 0. Define

P ~ \Pl> ) Pn—h $»)

by Pi = a< + ——7 for i = 1, • • • , n - 1, and P H = u - « = -1.

“ 7l — 1 —

— ♦ ...

P is an imputation (cf. (55:12) above). p„ < u, or equally (55 :L), excludes

-4

P from V.

THE SIMPLE GAME [!,•••,!,« - 2] h

481

Hence there exists an a in V with a H P . By (55 :L) ^ a* < Pi

for all t = 1, • • • , n — 1. But both (55:1) and (55:2) in (55: A) require —

since a H p — that > Pi for at least one i = 1, • • • , n — 1. Thus
we have a contradiction.

Note that now (55:12) becomes

(55 :N) "S a» = 1.

»-l

The essential results of this analysis are (55 :K), (55 :L), (55 :M). They
can be summarized as follows: 1

The worst possible outcome for the chief player is complete defeat
(value —1). There is one and only one arrangement — i.e. imputation —
(in V) which does this, and for all other players this is the best possible one
(in V).

This arrangement (in V) is the state of complete cooperation against
the chief player. 1

The reader will note that while this verbal formulation is not at all
complicated, it could only be established by a mathematical, not by a
verbal, procedure.

65.5. Case (II) : Determination of V
66 . 5 . 1 . We are now able to investigate V.

(55:0) Consider an imputation P = [Pi, • • • , p n ] with pi ^ a,

for some i = 1, •••,»- 1 and p n ^ a>. Then / 8 belongs
to V.

Proof: Assume that P does not belong to V. Then there exists an a
— ► — ►

in V with a H p. Hence (55:1) or (55:2) of (55:A) must hold. As
— ►

a is in V f an ^ w ^ P *, and this excludes (55:1). By (55 :L) a* g g* for all

i = 1, • • • , n — 1, hence a» ^ a* ^ Pi for at least one i = 1, • • , n — 1,

and this excludes (55:2). So we have a contradiction in both cases.

(55:P) oti — 2 co for i = 1, • • • , n — 1.

Proof: Assume that a< < n - 2 - w for a suitable i = 1, • • • , n - 1,

i.e. that

— (n — 2) + oti + w < 0.

1 All this applies, of course, to the case (II) only.

1 This expression was also used — in a related, but somewhat different sense — in the
last part of 55.2.

482 SIMPLE GAMES

Then we can choose ft ^ -1 (j = 1, • • • , n — 1, j ^ i, i.e. n — 2 values

of j ) pi ^ «*, 0n > w with ft- = 0, forming the imputation

i-i

0 = (ft,

> fin } •

This P meets the requirements of (55:0), hence it belongs to V. But this
necessitates p n ^ d> by the definition of that quantity — contradicting
Pn >

Now put

(55:13) a* = Min t «i n -i «». 1

Then (55 :P) states this:

(55:14) a+ ^ n — 2 — co . 2

Denote the set of all i(= 1, • • • , n — 1) with
(55:15) a» = a *

by S*. By its nature this set must have these two properties:

(55:Q) S* S (1, • • • , n — 1), S+ is not empty.

66.5.2. We continue:

(55 :R) a* = n — 2 —

(55 :S) V consists of these elements: a 1 where t rqns over all S#,

and where a 1 = {a\, • • , aj^, a‘ n | with

( a, = a* for j = i f

co

— 1

for j = n,
otherwise.

Proof of (55 :R) and (55 :S) : We begin by considering an element p of V.

— ► — 1 ,

If ft < for all i = 1, • • • , n — 1, then (55:2) gives a 0 H P since

a 0 = {a\, • • • of n -ii w}. As a 0 belongs to V by (55:K), so a 0 , P are
both in V — hence this is impossible. So

(55:16) ft ^ Qt% ^ a* for some i = 1 , • • * , n — 1 .

Necessarily

(55:17) ft ^ —1 for all j = 1, • • • , n - 1, j ^ i,

and since p is in V, so
(55:18)

Pn = W.

1 This time the minimum is formed with respect to a finite domain!
* Cf. however, (55 :R) below.

THE SIMPLE GAME [1, • • • , 1, n - 2]*

483

Now £ 0, = 0 and by (55:14) - (n — 2) + a* + <i g 0, hence > is
i-i

excluded from all inequalities of (55:16), (55:17). Hence «< = a„, i.e.
i belongs to S*. And

ft =

«• = «*
0)

-1

i.e. 0 = a 4 as defined above.
So we see:

for j = i,
for j = n,
otherwise,

(55:19) Every /3 of V is necessarily an a 1 with i in S*.

Now V is not empty, hence an a { in V (i in S*) exists. Consequently
— » # "

this a* is an imputation, hence a} = 0, i.e. — (n — 2) + a* + <i = 0.

>-i

This is equivalent to (55 :R).

Consider finally any i of £*. Since (55 :R) is true, we have
— (n — 2) + a* + w = 0.

Hence ^ a} = 0, i.e. a' is an imputation. But a\ = = a*, a 4 n = a>,

y-i

— i — ►

hence (55 :0) guarantees that a i belongs to V. And since a\ = w, a < is
even in V. I.e.

(55:20) Every a * with i in S+ is an imputation and belongs to V.

(55:19) and (55:20) together establish (55 :S). (55 :R) was demon-

strated above. Thus the proof is completed.

55 . 5 . 3 . The essential results of this analysis are (55 :R), (55 :S) together
with the introduction of the set S*. It is again possible to give a verbal
summary. 1

The best possible outcome for the chief player assigns to him a certain
value o). In order to achieve this he needs precisely one ally who can be
selected at will from a certain set S+ of players. This set consists of those
among the players 1, • • • , n — 1 who are least favojed in the state of
complete cooperation against the chief player, referred to at the end of 55.4.

Thus the arrangements which the players 1, • • • , n — 1 make between
themselves, when they combine to defeat the chief player completely,
determine his conduct in those cases where he achieves complete success.
This “interaction” between fundamentally different situations is worth

1 All this applies, of course, to the case (II) only.

484

SIMPLE GAMES

noting. 1 It is also of interest that the natural allies of the chief player, when
he aims at complete success, are the least favored members of a potential
absolute opposition against him. 2

The concluding remark of 55.4. concerning the contrast of formulation
and proof applies again.

55.6. Case (II) : a and S *

55 . 6 . We determined in 55.4., 55.5. the two parts V, V of V.* It is
therefore time to turn to the last remaining part of V:V*.

Let a be the set of all a with = g» = a* for all i in £*. Then we
have:

(55 :T) V u V*^a.

Proof : Consider an a in V u V*. We must prove a* = g» for all i in S*.
Now a.i £ ai for all i = 1, • • • , n — 1 by (55 :L). Hence we need
only exclude a, < a, when i is in S*.

For i in S+ form the a ' of (55 :S). It belongs to V, so a\ = co; a

belongs to V u V*, so a n < a>. Hence a\ > a„. Now ct % < a* means
— ^ ^

a\ = > «i, hence a by (55:1) — and this is impossible, since

a *, a both belong to V.

(55 :U) V £ G if and only if S+ is a one-element set or a^, = — 1;

otherwise V and d are disjunct.

Proof: Consider an a in V. Then a = a* (from (55 :S)), i in S*.

Comparing the definitions of a * and d makes it clear that this belongs to d
if and only if S+ has a unique element i or a* = —I.

1 In 4.3.3. we insisted on the influence exercised by the “ virtual’ ^ existence of an
imputation — i.e. of its belonging to a certain standard of behavior (solution) — on all
other imputations of the same standard. Almost all solutions of n ^ 3 — person games
which we found can be used to illustrate this principle. A specific reference to it was
made at an early stage of the discussion, in 25.2.2. The present instance, however, is
particularly striking.

* Political situations to illustrate this principle are well known and in connection with
them its general validity is frequently asserted. It is difficult to deny, however, that the
case which can be made purely verbally for this principle is no better than that which
could be made for a number of other conflicting ones.

The point is that for the particular game — i.e. social structure — we consider at
present, this and no other principle is valid. To establish it a mathematical proof of
some complexity was needed. All purely verbal plausibility arguments would have
been inconclusive and ambiguous.

•The set S * is still unknown, although restricted by (55:Q). The numbers
orit • • • , Qn-i are also unknown, but restricted by (55 :N). They determine a* (their
minimum), w, « are given by (55 :M), (55 :R). The determination of these unknowns
will be attended to later. Cf. (55:0 ; ) (i.e. (55:L'), (55:N') and (55 :P')).

Nevertheless, the form of V and of V has been found, and the remaining uncertain-
ties are of a less fundamental character.

THE SIMPLE GAME [!,•••,!,»- 2]*

485

The verbal meaning of (55 :T), (55 :U) is this: Each player of the least
favored group (S+, cf. the end of 55.5.) reaches his optimum 1 in every
apportionment in which the chief player is not fully successful (i.e. in
V u V*). When the chief player is fully defeated (i.e. in V), this is even
true for all players I, • • • , n — 1 (cf. the end of 55.4.). When the chief
player is fully successful then this is true for one and only one player, who
may be any member of the least favored group (<S*, cf. the end of 55.5.).

55.7. Cases (II') and (II"). Disposal of Case (IT)

55.7.1. Consider the case S+ = (1, • • • , n — 1), to be called case (IP).

In this case g* = a* for all i = 1 , • • • , n — 1, so (55 :N) gives (n — l)a* = 1 ,

i.e., a* = - y and (55 :R) gives d> = n — 2 — If « belongs to 0t

then a t = on = a* = — for i = 1, • • • , n — 1. Hence a n = — 1,
n — 1

— ► ( 1 1 )

i.e. a = I ~ _ i > ■ • • > - — ^ — 1 j • By (55:T) this is equally true for

all a in y u V*.

— * — *

This a is clearly the unique element a 0 of V by (55 :K), hence V* is

empty. Hence V = V u V, and now (55 :K), (55 :S) give
(55 :V) V consists of these elements:

<*> * * - (dr ' ■

(b)

where i = 1, • • • , n — 1 and where
a * = (a*„ • • • , aj,|

with

a

for j = i,

for j = n,
otherwise.

(55 :V) determines the only possible solution V in the case (IP) . This
does not necessarily imply, however, that this V is either a solution or in
case (IP). Indeed, if it failed to meet any one of these two requirements,
then we would only have shown — although in a rather indirect way — that

1 His individual optimum within the given standard — i.e. solution — V. For the
player i(— 1» • • , n — 1) this optimum (maximum) is g» owing to (55 :L) — although

gi was originally defined as his pessimum (minimum) in the part V of V.

486

SIMPLE GAMES

no solution in the case (IP) exists. We will prove, therefore, that both
requirements are met. 1

66 . 7 . 2 .

(55 :W) The V of (55 :V) is the unique solution in the case (IP).

Proof: We need only show that this V is a solution in the case (IP) —
the uniqueness then follows from the above, i.e. from (55:V).

Case (IP) is easily established: Clearly, for this V

w = —1. o> = n — 2 - — ->

n — 1

«i = ’ • ’ = 1 = «♦ = - i 9 s + = (!> * * * » n - 1).

It remains for us to prove that V is a solution, i.e. to verify (30:5:c)

in 30.1.1. To this end we must determine the imputations P which are
undominated by elements of V.

For a°H 0 (55:1) is excluded since a® = — 1. So this domination
can operate through (55:2) alone, hence it amounts to a® > ft, i.e. ft < - *

1

for i = 1,

,n- 1.

For a k H p , k = 1, ; • • , n — 1, (55:1) is excluded when i k
and (55:2) is excluded since a£ = — 1 for i k. So this domination can
operate through (55:1) with i = k alone, hence it amounts to a* > ft for

j = k , n, i.e. ft < — — ft < n - 2 - 1

1

n — 1

Hence p is undominated by elements of V if and only if this is true:
ft ^ z ~— r holds for some i = 1, • • • , n — 1 and it holds even for all

n — 1

these i in case that ft < n — 2 —

1

n — 1

Thus ft < n — 2 —

1

necessitates ft,

, ft-i £

1

Also

1 UVVVWtVMVVU ) f'n—l == „ 1

n — l n — l

ft ^ — 1 . Hence £ ft = 0 yields = for all these ^ relations, i.e. 0 = a °.

»-i

On the other hand ft ^ n — 2 —

•(- 1, *

1

necessitates Pi ^

for one

1 U VVUUUl VU WO

n — 1

, n — 1) and ft ^ — 1 for the other n — 2 values of j. Hence

£ ft = 0 again yields = for all these ^ relations, i.e. 0 = a *.

y-i

1 Cf. this situation with (55 :G), where the case (I) was settled. No such secondary
considerations were needed there, because (55 :G) was ab ovo necessary and sufficient.

THE SIMPLE GAME [1, ■ • • , 1, n - 2] h

487

So the 0 undominated by V are a 0 and a ', • • • , a n_1 ; i.e. precisely
the elements of V* as desired.

65 . 7 . 3 . This solution is of importance because it is a finite set — as we
shall see it is the only solution with that property. If the general coalition
against the chief player is formed, the n — 1 participants share in it equally

— as described by a °. If the chief player finds an ally, he gives him the

same amount as a 0 and retains the remainder — as described by
— > — >

a ', • • • , a n_1 .

All this is perfectly reasonable and non-discriminatory. 1 Nevertheless
this is not the only possible solution — we found another one in 55.3. (cf.
(55 :G)) and more will emerge in the sections which follow.

55.8. Case (II") : Ct and V'. Domination

66 . 8 . 1 . Consider next the case S* (1, • • • , n — 1), to be called
case (II").

Using (55 :Q) we may also formulate this as follows:

(55:X) <S* c (1, • • • , n — 1), S* not empty.

We can also say: Cases (II') and (II") are characterized respectively
by the absence and by the presence of discrimination within the possible
general coalition against the chief player.

As we enter upon the discussion of case (II") the following remark is
indicated :

The argumentation of 55.4.-55.7. was mathematical, but the (inter-
mediate) results which were obtained there allowed simple verbal formula-
tions. I.e. it was possible to work into the mathematical deduction
relatively frequent interruptions, giving verbal illustrations of the stages
attained successively.

This situation changes now, insofar as a longer mathematical deduction
is needed to carry us to the next point (in 55.12.) where a verbal interpreta-
tion is again appropriate.

55 . 8 . 2 . We now proceed to give this deduction.

Write V' = Ct n V (the part of V in ft). By (55 :T), (55:U) V' = V u V*
orV' = VuV*uV = V» according to whether the condition of (55 :U) is
not or is satisfied.

(55 :Y) The condition (30:5:c) holds for V' in Ct.

Proof: Replace (30:5:c) by the equivalent (30:5:a), (30:5:b) in 30.1.1.

Ad (30:5:a): Since V' £ V, elements of V' cannot dominate each other
because the same is true for V.

1 The special cases n — 3, 4 of this solution are familiar: For n — 3 it is the non-
discriminatory solution of the essential three-person game; for n » 4 it was discussed in
35.1.

488

SIMPLE GAMES

Ad (30:5:b): Let 0 in Q, not be in V'. Then we must find an a in V'
with a H 0.

To begin with, 0 is even not in V. Hence an a in V with a H 0
— ► . —

exists. This a must be in V' if it is not in v (cf. the remarks preceding
(55 :Y)), and this would establish our statement. So we need only to exclude

a from V.

Assume that a is in V, i.e. (by 55 :S) a = a k f k in S*. We have
— ► — ►

ot k ^ 0. (55:1) is excluded when i 9 * k, and (55:2) is excluded since

a* = —1 for i k (i = 1, • • • , n — 1). So this domination can only
operate through (55:1) with i = fc, hence it implies a* > 0*, i.e. 0* < Qk = a*.

However, this is impossible, since 0 belongs to Ot.

55 . 8 . 3 . Thus our task is now to find all solutions (i.e. all sets fulfilling
(30:5:c) in 30.1.1.) for Ct. This necessitates determining the nature of
domination in Ct.

(55 :Z) For a, 0 in G, a H 0 is equivalent to this: a n > 0« and

> fa for some i in (1, • • • , n — 1) — £*.

— > — >

Proof: For a H 0 (55:1) is excluded when i is in S*, and (5&:2) is
excluded since a* = 0 A ( = g* = a*) for all k of £*.

So this domination can only operate through (55:1) with i in (1, • • • ,
n — 1) — S*. And this means a n > 0 n and a t > 0*, as asserted.

We have replaced the set of all imputations by Ct and the concept of
domination described in (55 : A) by that one described in (55 :Z). Otherwise
the problem of finding all solutions has remained the same. The progress
is that the concept of domination in (55 :Z) can be worked more easily than
that one in (55: A) as will be seen in what follows.

56.9. Case (II") : Determination of V'

55 . 9 . 1 . Let p be the number of elements in £*.

Then we have:

(55:A') 1 ^ p ^ n — 2.

Proof: Immediate by (55 :X).

(55 :B')

Proof: — 1 2= a* is evident. Next a* = a* for i in <S„, a< > a* for i
in (!,•••, n — 1) — S*, and by (55 :A') neither set is empty. So

THE SIMPLE GAME [1, • • • , 1, n - 2]*

489

n - 1 j

y on > (n — l)a*, and hence (55:N) gives 1 > (n — 1)«*, a* <

. - 71—1

as desired.

An a in a has p fixed components: the a»( = = a *), t in £*; and
n — p variable ones: the a iy i in (1, • • • , n) — S+. These are subject
to the conditions

(55:21) a, £ -1 for i in (1, • • • , n) - S* f

and &% = 0, i.e.

»-i

(55:22) £ «< * -P“*-

» in ( 1 n) — 5#

The lower limits in (55:21) add up to less than the sum prescribed in
(55:22), i.e. — (n — p) < —pa*. Indeed, this means cr* < ? = - — 1.

p p

And by (55:A') p<n — 1, so-— 1> — — - — 1 = — , and (55:B')
J V n — 1 n — 1 v 7

guarantees a* <

So we see:

(55 :C') The domain G is (n — p — l)-dimensional.

65.9.2. We now proceed to a closer analysis of V' and of G. 1
Put

(55:23) w* = n — p — 1 — pa*.

By (55 :R) we can write

(55:24) a,* = — (p — l)(a* + 1).

(55:D') w* = w if and only if S* is a one-element set (i.e. p = 1) or

a* = — 1 i.e. if and only if the condition of (55 :U) is unfulfilled;
otherwise o>* < co.

Proof: Since p ^ 1, a* — 1 by (55 :A'), (55:B'), this is immediate
from (55:24).

(55 :E') Max- a,* = a>*.

a in ft

4 The lemmas (55:D0-(55:P0 which follow are the analytical equivalent of the
graphical deduction of 47.5.2.-47.5.4. The technical background differs, but the analo-
gies between the two proofs are nevertheless very marked — the interested reader may
follow them up step by step.

(55 :C0 shows that a graphical discussion would have to take place in a (n — p — 1)
dimensional space (by (55: AO this is £ 1, £ n - 2). This is the reason why we use an
analytical one. (The graphical proof referred to above took place in a plane, i.e. it
required 2 dimensions.)

490

SIMPLE GAMES

(55 :F') This maximum is assumed for precisely one a in Ot:

with

( a» = a* for i in S+,

co* for i — n,

— 1 otherwise. 1

Proof of (55 :E') and (55 :F'): It is clear from the definition of Q that for

the a of G the variable component a n assumes its maximum when the other
variable components — i in (1, • • • , n — 1) — *S* — assume their
minima. These minima are — 1. So for this maximum

oti = a*
-1

for i in S+,

for i in (1, • • • , n — 1) — S*.

n — 1

Now a n — — a * — ~~P a * + (n — 1 — p) — n — p — 1 — pa*. By

i — 1

(55:23) this means a n = w*.

This proves all our assertions.

(55 :G') a * belongs to V'.

Proof: a * belongs to a, for any a of a (55 :E'), (55 :F') give

«n ^ «n(= <*>*)•

So (55 :Z) excludes a H a *, and therefore (55 :Y) necessitates that a*
belong to V'.

55 . 9 . 3 . After these preparations the decisive part of the deduction
follows :

) ) — — >

(55 :H') If a, 0 belong to V', then a n = /3 n implies a = p .

Proof: Consider two a , ($ in V' with a n = 0 n .

Put 7 » = Min (a t , 0 t )(i = 1, • • • , n — 1, n) and assume first that

n n

£ Ys < 0, say £ 7. = * > 0.

*-1 t = l

— >

Put 6 = {5i, • • • , 8 n - 1 , 6 n ) where

7» fori in aS*,

7* + — * v for i in (1,
n — )) 1

n — 1 , n) — S*.

1 Comparison of this definition with (55 :D') shows that this a * is an a *, i in S — i.e.
that it belongs to V — if and only if the condition of (55 :U) is fulfilled.

Since a * belongs to a this is in agreement with the result of (55 :U).

THE SIMPLE GAME [1, • • • , 1 , n - 2],

491

This 5 is clearly an imputation, and as i in 5* gives 5, = 7 < =<*<=& =«<=<**,
— >

so 8 belongs to Ct. We have 8 n > y n = a n = ft», and for i in

— > — > — > — >

(1, * * • , n — 1) — S *, 5* > 7» = or ft, hence 5 h a or 8 H 0 .

Since a , 0 belong to V', this excludes 5 from V'. Hence there exists
— ► — > — >

an tj in V' with t\ H 5 .

Now by (55:Z) ry n > 5 n and rji > 8 t for an i in (1, • • • , n — 1) — S*.

— ► — >

A fortiori rj n > 8 n > y n = a n = ft», t/, > 5; > y< = <*» or ft. Thus ij H a
or i) H 0 . As a , p , r\ all belong to V', this is a contradiction.

n

Consequently ^ 7 » < 0 is impossible, so

i- 1

(55:25)

Now 7 i g a», 7 * ^ ft and ^ a» = ^ ft = 0 . Hence (55:25) yields = for

» = i » = i

— > — >

all these ^ relations, i.e. 7 » = ou = ft. This proves a = 0 as desired.

— >

( 55 : 1 ') The values of the a n for all a in V' make up precisely the

interval

— 1 ^ a n ^ a)*.

Proof: For an a in V', a n ^ —1 is evident, and a n S c*>* follows from
(55 :E'). Hence we need only exclude the existence of a y i in

— 1 ^ 2/i ^ co*,

such that a» i/i for all a in V'.

— > — >

There exist certainly elements a of V' with a n ^ y i: Indeed a * belongs
to V' by (55:G'), and a* = = 2 /i- Form

Min.

a in V' with a n ^ 2/1

«n = y 2,

1 In this case it is not necessary to form the exact minimum, but the procedure which
achieves this is somewhat longer than the one used below. That this minimum can be
formed, i.e. that it exists and is assumed, can be ascertained in the same way as in foot-
note 1 on p. 384. Cf. in particular (*) loc. cit. What is stated there for V is equally
true for the analogous set V' in a and for the intersection of V' with the closed set of

the a with an ^2/i.

Because of this need for closure we must use the condition a n ^ y i and not a n > y\
although we are really aiming at the latter. But the two will be seen to be equivalent
in the case under consideration. (Cf. (55:26) below.)

492

SIMPLE GAMES

and choose an a + in V' with a+ ^ 2/1 for which this minimum is assumed:

a + = y 2 . By (55 :H') this a + is unique.

So 2/2 ^ 2/i» and since necessarily a+ ^ y\ f so 2/2 5* yi, i.e.

(55:26)

l/i < 2/a-

It follows from the definition of 2/2 that

(55:27)

2 /i ^ a„ < 2/2 for n0 a V'.

Now put 2/1 = 2/2 — €, « > 0 and form the imputation

0 = {/Si, * * * , /Sn-l, ft»|

with ft = a+ — € = 2/2 — * = 2 / 1 ) ft = = «» = «* for i in ft, ft =

a+ H 7 for i in (1, • * • , n — 1) — ft. Clearly fi belongs to

n — 1 — p

— ^ ^

Gt and ft = t/i excludes /S from V'. Hence there exists a 7 in V' with

7 *"■ fi •

By (55:Z) this means 7 n > ft and 7» > ft for an z in (1, • • • , n — 1) — ft.
Now 7 n > ft = 2/1 necessitates by (55:27) 7n ^ ^2* 7n = 2/2 would

imply 7 = a + (by (55:H'), cf. above). Hence 7 , = a+ < ft for the
above z in ( 1 , • • • , n — 1 ) — ft, and not 7 , > ft as required. Hence
7 » > 2 / 2 .

Thus 7 n > 2/2 = and 7 »> ft > for the above z in (1, • • • , n — 1) — ft.

So 7 h a +, and as 7 , a 1 both belong to V' this is a contradiction.
66.9.4. By (55:1'), (55:H') we see: For every y in

— 1 ^ y ^ u*

there exists a unique a in V' with a n = y. Denote this a by

« ( y ) = \oti(y), * • • , c*n-i(2/), «n(y)}.

Clearly a n (y) = y and a t (y) = a» = a* for i in ft. So the functions which
matter are the a x {y) for z in (1, • • • , n — I) — ft.

Combining this with (55 :F) gives:

(55 :J') V' consists of these elements:

— ►

« (y)

where y runs over the interval

-1S^ «*,

THE SIMPLE GAME [1, • • • , 1, n - 2]» 493

and where a ( y ) = | or ,(?/), • • • , a n (j/)} with

/ oti = a* for i in S+,

otiiy) = < V . for i = n,

/ a suitable function of 2/ (and i) for i in
\ (1, • • • , n - 1) - S*.

55 . 9 . 5 . And to conclude :

(55 :K')

The functions ai(y), i in (1, • • • , n — 1)
fulfill the following conditions:

(55:K':a)

The domain of a,(y) is the interval

-1 £ y £ w*.

(55:K':b)

2/i ^ 2/2 implies a x {y ,) ^ <*,( 2 / 2 ).*

(55:K':c)

«.( — 1) =

(55:K':d)

a,(o*) = — 1.

(55:K':e)

X ai(y) = -pa* - i/. 2,5

» in (1, ••*,n — 1) — 5*

of (55 :J')

1 1.e. a,(y) is an antimonotonic function of y.

1 From these relations the continuity of all functions a*(l/), i in (1, • • • , n — 1) — S*
follows. Indeed, we can even prove more, the so-called Lipschitz condition :

(55:28) M 2 / 2 ) - «t(2/i)| ^ 1 2/2 - yi|.

Proof: This relation is symmetric in y 1 , y j hence we may assume yi ^ y 2 . Now
application of (55:K':e) to y - y x and y = yt and subtraction give

X M 2 / 1 ) - «<(yi)) = yt - y 1 .

» in (!,•••, n — 1) — <S*

By (55:K':b) all these addends a t (yi) — a»(2/*) are ^ 0, hence they are also ^ than their
sum yt — yi. Thus

0 £ a % (y 0 - a t {yt) £ yi - y 1 .

These inequalities make it also clear that the middle term is |a»(y*) — a*(y0| and
that the last term is |yt — yj. Hence we have

|«<(y*) - a<(»i)| ^

as desired.

The reader will note that we never assumed any continuity — we proved it! This is
quite interesting from the technical mathematical point of view.

* Note that (55:K':c), (55:K':d) do not conflict with (55:K':e). Indeed:

For y - -1 (55:K':e) gives ]j£ «<(-l) - -pa* 4- 1, hence

i in — 1) — S*

n — 1

(55:K':c) requires ^ — —pa* 4-1, a» - 1, agreeing with (55 :N).

i in — 1) — 5*

For y — w* (55:K':e) gives ^ a*(«*) = —pa* — «*, hence

i in (1,— ,n — 1) — S*

(55:K # :d) requires — (n — p — 1) — —pa* — w* , w* *»

(55:23).

n — p — 1 — pa*, agreeing with

494

SIMPLE GAMES

Proof: Ad (55:K':a): Contained in (55 :J').

Ad (55:K':b): Assume the opposite: j/i g j/ 2 and a,(j/i) < (for a

suitable i in (1, • • • , n — 1) — S t ). This excludes y x = y it so y i < y t .

Then a (j/j) H a (y x ), which is impossible since a (y i), a (j/ 2 ) both belong

to V'.

— ►

Ad (55:K':c): This is a restatement of the fact that a 0 belongs to V',
indeed it belongs to V. (Cf. (55 :K), (55 :M).)

— ►

Ad (55:K' :d): This is a restatement of the fact that a * belongs to V'
(cf. (55:G')).

^ n

Ad (55:K':e): a ( y ) is an imputation, hence ^ a 4 (y) = 0.

t- i

By (55 :J') this means that a 4 (t/> + p«* + y = 0,

* in (1 ,•••,« — 1) — S*

i.e. that £ a,(y) = —pa* — y> as desired.

i in (!,**•, n — 1) —

55.10. Disposal of Case (II")

65.10.1. The results obtained in 55.8.-55.9. contain a complete descrip-
tion of the solution V. Indeed: As we saw at the beginning of 55.8.2.
V = V' u V, although the addend V may be omitted (because it is £ V')
if and only if the condition of (55:11) is satisfied. V is described in (55:S),
V' in (55 :J')- These characterizations make use of the parameters

a. (t = 1, • • * , n - 1), a*, S*, u>, a>*,
a t (y)(i in (1, • • • , n - 1) - S*, -1 g y £ o*) f

which are subject to the restrictions stated in (55:N); (55:13), (55:15) in
55.5.1.; (55 :R); (55:23), (55:24) in 55.9.2.; (55:K').

Since this material is dispersed over seven sections, it is convenient to
restate the complete result in one place:

(55:1/)

(55:L':a) S* is a set c (1, • • • , n — 1), not empty. Let p be the

number of elements of S*, so that 1 ^ p g n — 2.

n — 1

(55:L':b) «i, • • • , g„-i are numbers ^ — 1, with £ a, = 1.

1-1

(55:L':c) For all i in S* a, = a*, for all i in (1, • • • , n — 1) — 5*

a, > a*.

Put o) = n — 2 — a*, a>* = n — p — 1 — pa*, so that
u - «* = (p - l)(a* + 1).

(55:L':d)

(55:L':e)

THE SIMPLE GAME [1, • • • , 1, n - 2]* 495

ai(y) is defined for i in (I, • • • , n — 1) — S*.

-1 g y g to*.

These functions satisfy the conditions (55:K':a)-(55:K':e).

V consists of these elements:

►

(a) a ( y ) where y runs over the interval — 1 ^ y ^ w*, and
where

with

« ( y ) = { «i(y), * • • , a n (y ) },

(y) =

= a* for i in S*,
y for i = n,

the a x (y) of (55:L':e) for i in
(1, • • • , n _ 1) -S*.

(b) a 4 where i runs over all S+ and where

with

for j = i,
for j = n,
otherwise.

Remark: If p = 1 (S* a one-element set) or a* = — 1, then w = w*
and the a 1 of (b) coincide with a {y) of (a) for y = a>*. If this is not the
case — i.e. p S 2, a* > — 1, then a> > w* and the a 1 of (b) are disjunct
from the a (y) of (a).

The reader will verify with little difficulty that all these statements
are nothing but reformulations of the results referred to above.

55.10.2. (55 :L ; ) must be followed by similar considerations as (55 :V).
We must investigate whether all V obtained from (55 :L') are solutions and
in the case (II")- Those of them which meet both these requirements
form the complete system of all solutions in the case (II")- We will prove
that all V of (55 :L') meet these requirements.

(55:M') The V of (55 :L/) are precisely all solutions in the case (II")-

Proof: We need only show that every V of (55 :L') is a solution in the
case (II") — that these V are precisely all such solutions then follows from
(55 :L').

Case (II") is easily established: Clearly for this V, w = — 1 , and

W, gl> ' • * > «n-l, S*

496

SIMPLE GAMES

(in the sense of their definitions given in 55.2.-55.5.) are precisely the
quantities designated in (55 :L') by these symbols, 1 hence

= (I, • • • , n — 1)

by (551/ :a).

It remains for us to prove that V is a solution. In the present case
we will do this by proving that V fulfills (30:5:a), (30:5:b) in 30.1.1.

Ad (30:5:a): Assume a H p for a , P in V. We must distinguish

to which cases (a), (b) of (55 :L') a , P belong. There are four possible
combinations:

a, P in (a): I.e. a = a (y i), p = a (y t ) and so a (yi) H a (y t ).
Now (55:1) is excluded when i is in <S* and (55:2) is excluded since
«»(j/i) = «.(j/») = «< = «* for i in <S*. So this domination can operate
through (55:1) with i in (1, • • • , n — 1) — S+ only. By (55:L':e) this
means a»( j/i) > oc n (yt), y\ > yt, and ou{yi) > «,(j/ 2 ) for a suitable i in
(1, • • • , n — 1) — <S„, contradicting (55:K':b).

a in (a), p in (b): I.e. a = a(y), P = a * (i in <S*), and so
— ► — ►

a (y) H a\ Now (55:1) is excluded, since a n (y) - y ^ w* ^ o) = o*,
and (55:2) is excluded, since oti(y) = a\ = a. = a*. So we have a con-
tradiction.

a in (b), P in (a): I.e. a = a' ( i in £*), P = a (y), and so

a i H a (y). Now a\ = a*(y) = a t = a* and for j i y n, a) = — 1 g a t (y),
i.e. a) S oLjiy) for all j = 1, • • • , n — 1. This excludes both (55:1),
(55:2), and gives a contradiction.

a , P in (b): I.e. a = a\ P = a k (i, k in £*), and so a * H a *.
Now a' n = a* = w, thus contradicting (55 :B)

Ad (30:5:b): Assume that P is undominated by the elements of V.

— >

We wish to prove that this implies that p belongs to V — which establishes
(30:5:b).

Assume first that p n <i. If ft < = a* for all i — 1 , • • • , n — 1 ,

then a( — 1 )h P, contradicting our assumption. Hence |8» ^ lor
some i = 1, •••,» — 1. Now the argument used in the proof of (55:R)

shows that necessarily i in S t and p = a*. Therefore ~P belongs to V
in this case.

obtains from (b), oi, • • • , g._i from (a) with y — — 1, and then a*, <8* from
(SSdL'as).

THE SIMPLE GAME [1, • • , 1, n - 2]» 497

Assume next that /3„ < u. If /3< < g, = a* for some i in then

— * — ^

clearly a i H contradicting our assumption. So ft ^ a» = a* for all
* in ft.

» n — 1

Now ft = 0 gives ft = — ]£ft^n — p — 1 — pa* = w*, i.e.

»-i t-i

-1 ^ ft ^ CO*. Put p = ft.

Assume that ft ^ a»(p) for all i in (1, • • • , n — 1) — ft. Then we

have clearly ft ^ a»(2/) for all i = 1, • • • , n. (For i in ft and i = n

n n

we have even = , cf. above.) Hence ft = £ «»(!/) = 0 necessitates

»-l i-1

that we have = in all these ^ relations. So 0 = a (y). Therefore 0
belongs to V in this subcase too.

There remains the possibility that ft < ai(y) for a suitable i in (1, • * * ,
n — 1) — ft. A sufficiently small increase of y (from y = ft* to some
p > ft) will not affect this relation ft < a*(p). 1 For this new i/ we have

y > ft, a,(t/) > ft, and therefore a (p) H 0 contradicting our assumption.
Thus all possibilities are accounted for.

65.11. Reformulation of the Complete Result

66.11.1. These three cases (I), (II'), (II") — into which we subdivided
our problem — have been completely settled by (55 :G), (55 :W), (55 :M')
respectively. Let us now see to what extent these three classes of solutions
are related to each other.

Among the undetermined parameters occurring in (55 :L') — i.e. in
(55:M'), describing case (II") — is the set ft. According to (55:L':a)
this is any set S (1, • • * , n — 1) with the exception of (1, • * • , n — 1)
and Q. This raises the question whether it is not possible to find some
interpretation for these excluded cases ft = (1, • • • , n — 1) and ft = ©
also.

For ft = (1, • • • , n — 1) the answer is easy. If w r e use this ft
(disregarding (55:L':a) to this extent), then we obtain (using all other parts

of (55 :L')): p = n — 1 by (55:L':a), on =

Qf»- 1 = Of*

1

71—1

by (55:L':b), (55:L':c), a = n - 2 - "* = b y (MiL':d).

There is no occasion to introduce the functions a»(y) of (55:L':e), since
(1, • • • , n — 1) — ft is empty. Inasmuch as the interval — 1 ^ y £ «*
plays a role (in (a) of (55:L':e)), it must be noted that it shrinks to the
point y = — 1 (since w* = —1). Now comparison with (55 :V) discloses
that under these conditions (55 :L') coincides with (55 :V).

1 a< (y) is continuous! Cf. footnote 2 on p. 493.

498

SIMPLE GAMES

So we have :

(55:N') If we include in (55 :L':a) S+ = (1, • • • , n — 1) (hence p =

n — 1) also, then (55 :L') enumerates all solutions in the cases
(IP) and (II") : Case (IP) corresponds to S+ = (1, • • • , n — 1)
and case (II") to S+ (1, • • • , n — 1).

55 . 11 . 2 . After this result one might feel inclined to correlate the remain-
ing exception S* = © with the remaining case (I). However, inspection
of[(55:L') with S* = © and comparison with (55 :G) show that this is not
possible — at least not in this direct way.

Indeed: Use of (55:L) with S* = © (hence p = 0) gives an empty (b),
so a V coinciding with (a) — i.e. V is the set of all

« (: y ) = { a i(y)t • • • , ttn-i(y), y],

— 1 ^ y ^ w*, with suitable functions ai(p), • • • , a„_i(y). Disregarding

other maladjustments 1 we note: In this arrangement the a„ of an a in V
determines its a h * • • , a n -u while in (55 :G) a„ was constant and

ai,**-, a n _i

arbitrary ! 2

Summing up:

(55:0') All solutions V are enumerated by (55 :G) — Case (I) — and

(55 :N')— Cases (IP) and (II"). (55:N') coincides with (55:L'),

when (55:L':a) is widened to include all £* Q (1, • • • , n — 1)
with £* ^ ©. The exclusion of S+ = © is necessary; this
choice would produce a V which is not the solution of (55 :G), and
indeed is no solution at all.

55 . 11 . 3 . We conclude with the following observations:

(55 :P')

(55:P':a) In case (IP), i.e. S* = (1, • • • , n — 1), p = n — 1, we have:

o )* = —1, i.e. the interval —1 ^ y ^ w* of (55 :L' :e) shrinks to

a point. Also a* = —

n — 1

(55:P':b) In Case (II"), i.e. S+ c (1, • • • , n — 1), p < n — I, we have

co* > — 1, i.e. the interval —1 g y ^ o* of (55:L':e) does not

shrink to a point. Also a* < — i— r-

1 Owing to p » 0 (55:23) now gives « - w* - -(a* + 1), hence we may have
> w, and so Max-» a* - Max_j y - although it should be «!

For ^ ©, (55:L':b), (55:L':c) gave Min^ t ; for £ - © they

give Min^j n _ x > a*, although the former was the definition of a« I

* The V of (55 :L') with 8 ■» © is thus not a set from our list of solutions, hence it is
no solution at all. It would have been easy to verify this directly.

THE SIMPLE GAME [1, • • • , 1, n - 2]» 499

Proof: Ad (55:P':a): We proved these statements immediately preceding
(55 :N')-

Ad (55:P':b): We saw in the proof of (55 :B') that a# < -> hence

P

w* + 1 = n — p — p a * > 0, 03* > — 1. a* < — -i— r was stated in (55:B').

Tl A

55.12. Interpretation of the Result

65 . 12 . 1 . We can now begin to interpret this result. It is hardly possible
to do this in an exhaustive way for two reasons. First the final result —
contained in (55:0'), i.e. in (55:G), (55:K'), (55:L') — is rather involved,
hence a precise statement must necessarily be mathematical and not verbal.
Any verbal formulation would fail to do justice to some of the numerous
nuances expressed by the mathematical result. Second we still lack the
necessary experience and perspective for a really thoroughgoing interpreta-
tion of a situation like the present one. The game which we consider
here is a characteristic n- person game in some significant ways, as we set
forth in 54.1.2. and 54.3. But our success in determining all of its solutions
is still an isolated occurrence (the case of 54.2.1. notwithstanding). It will
take many more discussions like this one before one can attempt really
exhaustive interpretations of characteristic n- person games.

It is nevertheless useful to do a certain amount of interpreting — without
any claim of completeness. We have seen in several previous instances
that such interpretations give valuable guidance for the further progress
of the theory. Besides, this procedure does throw some light on the
significance of our rather complicated mathematical result.

Since we do not try to be complete, the interpretation is best made
in the form of several remarks.

65 . 12 . 2 . First: The solution of Case (I) described in (55 :G) is an infinite
set of imputations. The same is true for the solutions of (II"), described
in (55 :L') (cf. (55 :N')) since the y mentioned there varies over an entire
interval which does not shrink to a point. (Cf. (55:P':b).) On the other
hand the solution of Case (II') is a finite set of imputations as was already
observed at the end of 55.7. 1 This solution also has the attractive property
of sharing the full symmetry of the game — i.e. invariance under all permuta-
tions of the players 1 , * • • , n — 1 .

Thus it is in several ways the simplest solution of our game. Heuristic
discussions of its special cases n = 3, 4 (in 22., 35.1. respectively) led to this
solution and it is easy to extend them to the general n. 2 It takes the full
machinery of our formal theory to find the other solutions.

It will be sufficiently clear to the reader by now that these other solutions
can in no way be disregarded. Besides, the existence and the uniqueness

1 The reader may compare to the same effect (55:P':a), (55:P':b).

1 The (heuristic) argument would run as follows: The chief player needs an ally to
win, with any such ally he obtains n — 2. Thus if he wishes to retain the amount « (this
corresponds to the w of our exact deductions) he can concede each ally n — 2 — If

600 SIMPLE GAMES

of a finite solution is a favorable contingency in the present game, but by no
means general. 1

66 . 12 . 3 . Second: The above solution corresponded to the largest possible

:(1, • • • , n — 1). The other extreme is the solution which we associ-
ated with = © (cf. preceding (55:0'))* This is the solution in Case (I)
described in (55 :G). Like that one in the preceding remark it possesses
the full symmetry of the game. Indeed these two — the Cases (I) and (II') —
are the only ones with this symmetry. 2

On the other hand this solution is infinite. As we saw in 55.3. it expresses
the organizational principle that the chief player is segregated in the game
in the sense of 33.1. Inspection of (55 :G) discloses that this standard of
behavior — i.e. solution offers absolutely no principle of division among the
other players — i.e. all imputations where the chief player receives the pre-
scribed amount belong to it. This is perfectly reasonable by common
sense: The chief player being excluded, the other players can only combine
with eaot^ other unanimously. All quantitative checks in their relation-
ships (i.e. the possibility of siding with the chief player) being forbidden,
there is no telling what the outcome of their bargaining with each other
will be.

66 . 12 . 4 . Third: The remaining solutions are those in Case (II"), described
in (55 :L') (cf. (55 :N')), i.e. those with S* ^ ©, (1, • * * , n — 1). They
form a more complicated group than the two solutions dealt with above.
Indeed, they took up a considerable part — and the most involved one — of
our mathematical deductions. Their interpretation, too, is more difficult
and complicated. We will indicate the main points only.

We described in (55 :L') in detail how in all imputations of a standard
behavior — i.e. a solution — of this category the players of (1, • • • , n— 1)— S+

his n — 1 potential allies together can make more than that, i.e. if

(n - l)(n - 2 - «) < 1,

then his chances of finding an ally are destroyed — and this is the only limit to his
exactions.

Thus o> is only limited by (n — l)(n — 2 — «) ^ 1, i.e. <*> £ n — 2 — ^~Z T \ '

So « « n — 2 — — — : •
n — 1

So the chief player obtains n — 2 — ~Z~\’ ^ ^e succee d 8 in forming a coalition, and

of course — 1 ifhe does not. For the other players the corresponding amounts are “““j
and —1.

The reader can now verify that this is just the solution arrived at in (55 :V), i.e.
Case (IT).

1 As to the uncertainty concerning the existence cf. the end of the second remark in
53.2.2. An instance where the uniqueness fails is analyzed in 38.3.1.

1 Any other solution belongs to Case (II") and so has an S* * ©, (1, • • • , n - 1).
Hence an appropriate permutation of the players 1, • • • , n — 1 will carry an element
of into one outside, thus changing 5* and with it the solution under consideration.

THE SIMPLE GAME [1, • • • , 1, » - 2] h

501

are causally linked to the chief player. I.e. how the respective amounts
which they get are uniquely determined by the amount assigned to the chief
player. This connection was expressed by definite functions. 1 These
functions could be chosen in different ways, thus yielding different standards
of behavior — i.e. solutions — but a definite standard meant a definite choice
of these functions. Thus the uncor relatedness of the players 1, • • • ,n — 1,
so prominent in the second remark, is now gone. There is obviously some
kind of indefinite bargaining going on between the chief player and those
of (1, • • • , n — 1) — S*, 2 but the relationship of the latter players to
each other is completely determined by the standard.

It is worth while to emphasize once more this difference between the
situation described in the second remark and in the present one — i.e. between
the Cases (I) and (II")* In the former case there was bargaining between
all players except the chief player with absolutely no rules or correlations
to cover it, 3 so that the standard of behavior had to make no provision in
this respect. Now we have bargaining between the chief player and some
of the others, but this time the standard must provide definite correlations
and rules for the opponents of the chief player. Accordingly there is a
multiplicity of possible standards.

The qualitative types of indefiniteness arising in the Cases (I) and (II") ,
as discussed above, are a more general form of that one which we investi-
gated in 47.8., 47.9. The remarks made thereabout the 2-dimensional
(area) and one-dimensional (curve) parts of those solutions are indeed
applicable to our present Cases (I) and (II"), respectively.

While it is possible to motivate this difference by verbal arguments of
some plausibility, they are all far from convincing. The mathematical
deduction alone, such as we gave it, gives the real reason — and its relative
complication shows how difficult it must be to translate it into ordinary
language. This is another instance of a result which can be expressed, but
scarcely demonstrated, verbally.

55 . 12 . 6 . Fourth: The situation of the remaining players — those in £* —
has also its interesting aspects.

Inspection of (55 :L') shows that in every imputation of our solution
either all these players get the amount a* , or one of them gets a* and the
others the amounts - 1. From this one infers immediately:

(a) If S * is a one-element set then the player in S* gets always the
same amount : a*

(b) If a* = - 1 then each player of S * always gets the same amount:
- 1 .

(c) If neither of (a) or (b) is the case — i.e. if the condition of (55 :U)
(also referred to in (55 :D')) is fulfilled— then each player in S*

1 The «<(y), t in (1, •••,» — 1) — »S*.

* This corresponds to the variability of y in (55:L':e). Cf. also (55:P':b).

• Except for the assignment to the chief player who is segregated.

502

SIMPLE GAMES

always gets one of the two different amounts a* and — 1, and
neither can be omitted. 1

From these we can draw the following interpretative conclusions:

(d) In the two cases (a) and (b), but not in (c), the players of S+ are
segregated in the sense of 33.1.

(e) The case (a) where S * is a one-element set: S+ = (i), i = 1, • • • ,
ft — 1, expresses the segregation of the player i alone. The value
a * which is then assigned to him, is limited by (55 :B'):

(55:29)

This is a satisfactory complement to the segregation of the chief
player, Case (I), described in the second remark. 2 The value w
which was then assigned to the chief player was limited by (55 :G):

(55:30) — 1 < os < n — 2 —

(f) If S* is not a one-element set, then there is within the cases (a), (b)
only the possibility (b): a* = —1.

In other words:

If more than one player is to be segregated, then their set must not
contain the chief player, nor all other players, and the segregated
players must all be assigned the value:

(55:31) a* = -1.

(g) We conclude from (e), (f) that those sets of players which can be
segregated are precisely the sets of L 3 — the defeated sets.

(h) If only one player is segregated, then (e) shows that he need not be

discriminated against in an absolutely disadvantageous way. I.e. he
may be assigned more than — 1 . (55 :29) , (55 :30) also state the upper

limit of what this assignment can be: It is clearly the same amount
which this player would get in the finite solution of Case (I), dis-
cussed in the first remark. 4 It is very satisfactory that this extends
the result of 33.1.2. from ft = 3 to all ft.

(i) If, on the other hand, more than one player is segregated 6 then
(55:31) shows that there can be no concessions: They must all be
given the absolute minimum — 1.

1 I.e. both occur in appropriate imputations of the solution.

* This resolves the difficulty pointed out in footnote 3 on p. 475.

8 This is best verified by recalling the enumeration of the elements of W and so of L —
in the case C n -\ in 52.3.

4 n — 2 — n _ ' i f° r the chief player, - for the others. The assignment must be
less than these amounts.

5 I.e. the number p of elements in S+ is 2. Since p ^ n — 2 (cf. (55:L:a)) this can
happen only when n — 2 ^ 2 i.e. n £ 4. This is the reason why the phenomena of
(i) and (j) were not observed in the discussion of n - 3.

THE SIMPLE GAME [1, • • • , 1, n - 2]*

503

(j) This assertion must be qualified to the following extent: If S+ has
more than one element, the a* of (55:29) are still all possible —
indeed (55 :L') with (55 :B') allows for them explicitly. But the
situation of the players in S* is then described by (c), and can no
longer be termed segregation: They may join coalitions and thereby
improve their status.

It is clear that these remarks, particularly (g), (h), (i), invite further
comment. However we will now restrict ourselves to these indications
and return to the subject at another occasion.

55 . 12 . 6 . Fifth: We found a great number of solutions, characterized
by numerous parameters, some of which were even functions which could
be chosen with considerable freedom. The main classification, however,
was rather simple: It was affected by the set £*£(1, • • • , n — l). 1
The pairs S*, — S* exhaust obviously all partitions of / = (1, • • • , n)
into two sets. Possibly this is the first indication of a general principle.
In a simple game a partition into two complements seems to decide every-
thing, since one of them is necessarily winning and the other necessarily
defeated. In general games partitions into more sets may be equally
important. At any rate the role of S * in the present special case gives the
first idea of what may be a general classifying principle in all games.

We are not in a position, as yet, to give this surmise a more precise form.

1 We use as in the second remark S* — © to symbolize the case (I), the discussion
preceding (55:0') notwithstanding.

CHAPTER XI

GENERAL NON-ZERO-SUM GAMES

66. Extension of the Theory
56.1. Formulation of the Problem

66 . 1 . 1 . Our considerations have reached the stage at which it is possible
to drop the zero-sum restriction for games. We have already relaxed this
condition once to the extent of considering constant-sum games — with a
sum different from zero. But this was not a really significant extension
of the zero-sum case since these games were related to it by the isomorphism
of strategic equivalence (cf. 42.1. and 42.2.). We now propose to go the
whole way and abandon all restrictions concerning the sum.

We pointed out before that the zero-sum restriction weakens the con-
nection between games and economic problems quite considerably. 1 Spe-
cifically, it emphasizes the problem of apportionment to the detriment of
problems of “productivity ” proper (cf. 4.2.1., particularly footnote 2 on
p. 34; also 5.2.1.). This is especially clear in the case of the one-person game:
behavior in this situation is manifestly a matter of production alone, with
no conceivable imputation (apportionment) between players. And indeed
the one-person game offers no problem at all in the zero-sum case, and a
perfectly good maximum problem in the non-zero-sum case (cf. 12.2.1.).

Accordingly our present program of extending the theory to all non-zero-
sum games must be expected to bring us into closer contact with questions
of the familiar economic type. In the discussions which follow, the reader
will soon observe a change in the trend of the illustrative examples and of
the interpretations: we shall begin to deal with questions of bilateral
monopoly, oligopoly, markets, etc.

66 . 1 . 2 . Complete abandonment of the zero-sum restriction for our games
means, as was pointed out in 42.1., that the functions 3Cjt(ri, • • • , r n )
which characterized it in the sense of 11.2.3. are now entirely unrestricted.
I.e., that the requirement

(56:1) £ ac*(r, f • • • , t.) = 0

*-l

1 It should be noted that zero-sum games not only cover the type of games played for
entertainment (cf. 5.2.1.), but also that many of them describe quite adequately relation-
ships of a definitely social nature. The reader who has progressed up to this point and
recalls the interpretations which we have made in numerous cases will be fully aware of
the validity of this statement.

Thus the distinction between zero-sum and non-zero-sum games reflects to a certain
extent the distinction between purely social and social-economic questions. (The next
sentence in the text expresses the same idea.)

604

EXTENSION OF THE THEORY 505

of 11.4. and 25.1.3. is dropped with nothing to take its place. Accordingly
we proceed on this basis from now on.

This change necessitates a complete reconsideration of our theory
with all the attendant concepts on which it is based. Characteristic func-
tions, domination, solutions, — all these concepts are no longer defined when
(56:1) is dropped. We emphasize the fact that the problem which arises
here is a conceptual one, and not merely technical as were all those treated
in Chapters VI-X, on the basis of our theory of the zero-sum games. 1

56 . 1 . 3 . The prospect of having to start all over again would be very
discouraging: we have already spent considerable effort on these concepts
and the theory based on them. Furthermore we face a conceptual problem
and the qualitative principles on which our theory was based do not seem
to carry beyond the zero-sum case. Thus this final generalization — the
passage from the zero-sum to the non-zero-sum case — would seem to
nullify all our past efforts. We must find therefore a way to avoid this
difficulty.

At this point one might recall the comparable situation which arose in
42.2. There our transition from the zero-sum case to the constant-sum case
threatened — on a narrower scale — with similar consequences. They were
avoided by an appropriate use of the isomorphisms of strategic equivalence,
as effected in 42.3. and 42.4.

The usefulness of this particular device was, however, exhausted by
the application referred to: strategic equivalences extend the family of all
zero-games precisely to the family of all constant-sum games and no further.
(This should be clear from the considerations of 42.2.2., 42.2.3. or 42.3.1.)

So we must find some other procedure to link the theory of the non-zero-
sum games to the established theory of the zero-sum games.

66.2. The Fictitious Player. The Zero-sum Extension r

56 . 2 . 1 . Before going further we have to clarify a point of terminology.
The games which we shall now consider are those where — as stated in 56.1.2.
— condition (56:1) is dropped without anything else taking its place. We
talked of these as non-zero-sum games, but it is important to realize that
this expression is meant in the neutral sense, — i.e. that we do not wish to
exclude those games for which (56:1) happens to be true. It is therefore
preferable to use a less negative name for these games. Accordingly we
shall call the games with entirely unrestricted 5C*(ri, • * * , t*) general games .*

We have formulated the program of linking the theory of the general
games in some way to the theory of the zero-sum games. It will actually
be possible to do more: any given general game can be re-interpreted as a
zero-sum game.

1 Among these technical problems was one which we preferred to treat by a method
involving a certain conceptual generalization: the case of the constant-sum games, which
will be referred to further below in the text.

1 This is in agreement with 12.1.2.

606

GENERAL NON-ZERO-SUM GAMES

This may seem paradoxical since the general games form a much more
extensive family than the zero-sum games. However, our procedure will
be to interpret an n-person general game as an n + 1 -person zero-sum game.
Thus the restriction caused by the passage from general games to zero-sum
games will be compensated for — indeed made possible — by the extension
due to the increase in the number of participants. 1

66 . 2 . 2 . The procedure by which a given general n-person game is re-inter-
preted as an n + 1-person zero-sum game is a very simple and natural one.

It consists of introducing a — fictitious — n + 1-st player who is assumed
to lose the amount which the totality of the other n — real — players wins
and vice versa . He must, of course, have no direct influence on the course
of the game.

Let us express this mathematically : Consider the general n-person game
T of the players 1, * • • , n, with the functions 3C*(ri, • • • , r n ) (k =
1, • • • , n) in the sense of 11.2.3. We introduce the fictitious player n + 1
by defining

n

(56:2) 3C„ +l (r,, • • • , r„) = - £ 3C*(n, • • • , r„).

Jb-1

The variables n, • • • , r n are controlled by the — real — players 1, • • • , n,
respectively. They represent their influence on the course of the game.
Since it is intended that the fictitious player have no influence on the
course of the game, a variable r n+ i which he controls, was not introduced. 2

In this way we obtain a zero-sum n + 1-person game, the zero-sum
extension of T, to be denoted by I\

56.3. Questions Concerning the Character of I s

56 . 3 . 1 . In stating that we have re-interpreted the general n-person game
T as the zero-sum n + 1-person game T, we imply prima facie that the
entire theory of T has validity for T. This assertion requires, of course,
closest scrutiny.

We shall now undertake this investigation. It must be understood
that this cannot be a purely mathematical analysis, like the analyses in the
preceding chapters which were based on a definite theory. We are analyz-
ing once more the foundations of a proposed theory. Consequently the
analysis must in the main be in the nature of plausibility arguments, — even
if intermixed with subsidiary mathematical considerations. The situation
is exactly the same as in those earlier instances where we made our decisions

1 This may serve as a further illustration of the principle stated repeatedly that any
increase in the number of participants necessarily entails a generalization and complica-
tion of the structural possibilities of the game.

*The formalism of 11.2.3. provided a variable r* for every player k. (In order to
replace it for the present case we must replace its n by our n -f 1.) Hence one might
insist on the appearance of the variable r n +i of the fictitious player n 1.

This requirement is easy to meet. It suffices to introduce a variable r n +i with only
one possible value (i.e. to put p n +i - 1, loc. cit.). Actually one could even use any
domain of t *+ i (i.e. any 0»+i) as long as all 3C*(n, • • • , r„, r„+i) are independent of r n + i, —
so that they are really functions 3C*(n, • • • , r») as used in the text.

EXTENSION OF THE THEORY

507

concerning the theories of the zero-sura two-, three-, n- person games.
(Cf. 14.1.-14.5., 17.1.-17.9. for the zero-sum two-person game; Chap. V
for the zero-sum three-person game; 29., 30.1., 30.2. for the zero-sum
n-person game. For the general n- person game — i.e. the relationship
between the theories of T and r — the equivalent sections begin with 56.2.,
and go on to 56.12.)

The result of our analysis will be that it is not the entire theory of T —
as a zero-sum n + 1-person game, in the sense of 30.1.1. — which applies to
T, but only a part of it which we shall determine. In other words, we shall
find that not the system of all solutions for T, but only a certain sub-
system produces what will be interpreted as the solutions of T.

66 . 3 . 2 . The fictitious player was introduced as a mathematical device
to make the sum of the amounts obtained by the players equal to zero. It
is therefore absolutely essential that he should have no influence whatever
on the course of the game. This principle was duly observed in the defini-
tion of T as given in 56.2.2. We must nevertheless put to ourselves the
question whether the fictitious player is absolutely excluded from all
transactions connected with the game.

This caveat is not at all superfluous. As soon as T involves three or
more persons 1 the game is ruled by coalitions, as we observed at an early
stage of our analysis. A participation of the fictitious player in any coali-
tion — which is likely to involve the payment of compensations between the
participants — would be completely contrary to the spirit in which he is
introduced. Specifically: the fictitious player is no player at all, but only
a formal device for a formal purpose. As long as he takes no part in the
game in any direct or indirect form, this is permissible. But as soon as he
begins to interfere, his introduction into the game — i.e. the passage from
T to T — ceases to be legitimate. That is, F cannot then be regarded as an
equivalent, or a re-interpretation of F, since the real players of T, 1, * • • , n,
may have to provide against dangers or may profit by possibilities which
certainly do not exist in F.

66 . 3 . 3 . One might think that this objection is void due to the way in
which the fictitious player was introduced. Indeed, the amounts

no ... no

uvi, ,

which the real players 1, • • • , n obtain at the end of the play, do not
depend on any variable which he controls 2 — i.e. he has no moves in the play.
How can he then be a desirable partner in a coalition?

It may appear at first that this argument has some merit. The condi-
tions described make it seem that any coalition of real players is just as well

1 I.e. when n + 1 ^ 3 which means n ^ 2. Thus only the general one-person game
is free from the objections which follow. This is in harmony with the fact which we have
emphasized repeatedly, that the general n- person game is a pure maximum problem only
when n « 1.

n

2 Nor does the amount 3C n+l « — ^ 0C* which he obtains.

608

GENERAL NON-ZERO-SUM GAMES

off without the fictitious player as with him. Is he anything but a dummy?
If this were so, the theory of T could be applied without any further quali-
fications to T. However this is not the case.

It is true that the fictitious player, having no moves to influence the
course of the game, is not a desirable partner for any coalition. I.e. no
player or group of players will pay a (positive) compensation for his cooper-
ation. However he himself may have an interest in finding allies. The
amount which he gets at the end of the play — 3C n +i( T i, • • • , r«) — depends
on the moves of the other players — on n, • • • , r„ — and it may be worth
his while to pay one or more among the players a (positive) compensation
for ceasing to cooperate with the others. It is important not to misunder-
stand this: As long as T is played, i.e. as long as the fictitious player is
really a formalistic fiction, no such thing will happen; but if the game
really played is T, i.e. if the fictitious player behaves as a real player would
in his position, then his offer of compensations to the others must be expected.

66.3.4. As soon as the fictitious player begins to offer compensations
to other players for cooperation with him — which, as we saw above, amounts
to their non-cooperation with others — he is an influence to be reckoned with.
He offers to join coalitions and to pay a price for this privilege and his
willingness to pay is fully as good as a direct influence on the game exercised
by ability to make significant moves.

Thus the fictitious player gets into the game in spite of his inability
to influence its course directly by moves of his own. Indeed it is just this
impotence which determines his policy of offering compensations to others,
and thus sets the above mechanism into motion.

For a better understanding of the situation it may be helpful to give a
specific example.

56.4. Limitations of the Use of T

66.4.1. Consider a general two-person game in which each of the players
1,2 if left to himself can secure for himself only the amount —1, while the
two together can secure the amount 1. It is easy to specify definite rules
for a game to bring this about. 1 A particularly simple combinatorial
arrangement which does it, is as follows: 2

Each player will, by a personal move, choose one of the numbers 1,2.
Each one makes his choice uninformed about the choice of the other player.

After this the payments will be made as follows: if both players have
chosen the number 1, each gets the amount £, otherwise each gets the
amount —1.*

1 Thus it will be seen in 60.2., 61.2., 61.3. that the bilateral monopoly corresponds to
just this.

*This construction should be compared with the one used in defining the simple
majority game of three persons in 21.1., with which it has a certain similarity.

* With the notations of all 11.2.3.: 0i * — 2 and

5C,(r,, r,) - 3C,(t,, r.) “ { , ^thcnrisT ^

EXTENSION OF THE THEORY

509

It is easy to verify that this game possesses the desired properties.

Let us now consider the fictitious player 3 and form the game as defined
in 56.2.2., with its characteristic function v(S), (1,2,3). According

to what we said above

Obviously

v((l)) = v((2)) = -1,
v((l,2)) - 1.

v(e) = o,

and by the general properties of the characteristic function (of a zero-sum
game)

*((3)) - -v((l,2)) - -1,
v((l,3)) = -v((2)) = 1,
v((2,3)) = -v((l)) = 1,
v((l,2,3)) - -v(©) - 0.

Summing up:

1 0

when S has

1
0

0

1

elements.

2

3

This formula (56:3) is precisely the (29:1) of 29.1.2.; i.e.T is the essential
zero-sum three-person game in its reduced form, with 7 = 1. Thus it
coincides equally with the simple majority game of three persons, which was
discussed in 21. 1

Now we learned previously from the heuristic discussions of 21.-23. that
this game is nothing but a competition of all players for coalitions. Indeed,
this is immediately obvious, considering the nature of the simple majority
game of three persons (cf. 21.2.1.). Hence a fictitious player will certainly
show a strong tendency to enter into coalitions. In fact the game T is, as
far as the characteristic function is concerned, completely symmetric with
respect to its three players. Consequently the two real players 1,2 play
exactly the same role as the fictitious player 3, and so there is no reason
why their ability to enter coalitions should be at all different from his.*

56 . 4 . 2 . We can also revert to the argument used in the last part of 56.3.3.
and apply it to this game: If the fictitious player 3 in T behaves as a real one
would, he has every reason to try to prevent the formation of a couple of the
players 1,2, since he loses the amount — 1 if this couple is formed, and wins

1 Of course all these games coincide only as far as their characteristic functions are
concerned, but the entire theory of 30.1.1. is based on the characteristic functions alone.

1 To avoid misunderstandings we re-emphasife this: The rules of the game T, fully
expressed by the 3C*, are not at all symmetric with respect to the players 1,2,3; 3C * depends
on n, r* but not on n. It is only the characteristic function v(<S), S £ (1,2,3), which is
symmetric in 1,2,3. But we know that v(£) alone matters. (Cf. footnote 1, above.)

510

GENERAL NON-ZERO-SUM GAMES

the amount 2 if it is not formed. 1 Hence he will offer player 1 or player 2 a
compensation for disrupting this couple, i.e. for choosing r x or r*, respec-
tively, equal to 2 instead of 1. This compensation can be determined by
the considerations of 22., 23., and turns out to be $. 2 The reader may verify
this for himself, together with the fact that this procedure leads to the known
results concerning the simple majority game of three persons.

56 . 4 . 3 . The example of 56.4.1. gives substance to the objection formu-
lated in 56.3.3. and 56.3.4. Thus the fictitious player n + 1 can influence
the game T not directly through personal moves but indirectly by offering
compensations and thereby modifying the conditions and the outcome
of the competition for coalitions. As pointed out at the end of 56.3.3., this
does not mean that any such thing happens in r, i.e. as long as the fictitious
player is a mere formalistic fiction. It does happen in T if the theory of
30.1.1. is applied to it literally, — i.e. if the fictitious player is permitted to
behave (in offering compensations) as if he were a real one. In other
words, the considerations of the last paragraphs do not mean that we want
to attribute to the fictitious player abilities conflicting with the spirit in
which he was introduced. They served only to show that an uncompromis-
ing application of our original theory to T brings us into such a conflict.
Hence we must conclude that the zero-sum game T cannot be considered an
unqualified equivalent of the general game r.

What are we then to do? In order to answer this question, it is best to
return to the analysis of the specific example of 56.4.1. wherje the difficulty
was expressed fully.

56.5. The Two Possible Procedures

66 . 5 . 1 . One might try to escape from our present difficulty by observing
that it was brought about in 56.4.1. by the exclusive use of the characteristic
function. Indeed the game V there coincided with the simple majority
game of three persons — where the mechanism of coalition formation is
beyond doubt — only to the extent that they had the same characteristic
functions, but not the sameSC* (cf. in particular footnotes 1 and 2 on p. 509)
Thus a possible expedient might be to abandon the claim that the charac-
teristic function alone matters, and to base the theory on the JC* themselves.

At closer inspection, however, this suggestion appears to be entirely
without merit — at least for the problem under consideration.

First: Abandonment of the characteristic function v(£) in favor of the
underlying 3C* would deprive us of all means to handle the problem. For

1 By footnote 3 on p. 508 and by (56:2):

3C,(r,, r,) - — 3Cx(n, r,) - K,(r,, r.) - { 2 1 T ’ " ^

* This is the compensation which brings up the player 1 or 2 (who joined the fictitious
player 3) from the loss — 1, to a gain £ which is what he would obtain in a couple, i.e. in a
coalition of the players 1 and 2. It also brings the fictitious player's gain from 2 down
to J, which is indeed what it should be.

EXTENSION OF THE THEORY

511

zero-sum games we possess no general theory other than that of 30.1.1.,
based on v(S) exclusively. Thus the adoption of this program would make
our passage from the general game r to the zero-sum game T entirely
useless, since it would render us just as incapable of handling zero-sum
games as, originally, general games. Hence this sacrifice of our entire
existing theory would be reasonable only if it were quite certain that despite
its adequacy in all other respects there was no other avenue of escape.
However, neither of these two conditions is fulfilled.

Second: The retrogression from the characteristic function to the under-
lying 3C* does not meet the objections of the previous paragraphs. Indeed,
at the end of 56.3.2. as well as in 56.4.2. we did operate in a manner which
took the 3Cjt into account. We established the necessity for the fictitious
player in T to offer compensations in a direct manner, — a necessity which
was in no way dependent upon a replacement of T by a different game with
the same characteristic function. 1

Third: It will appear from the discussion which follows that it is not
necessary to sacrifice the theory based on the characteristic function, but
rather that the objections can be met by a simple restriction of its scope.

66 . 6 . 2 . Reconsideration of 56.3.2.-56.4.2. shows that we were not
justified in placing the blame for our present difficulties, with regard to the
behavior of the fictitious player, entirely upon the theory of 30.1.1.

The considerations of 56.3.2.-56.3.4. and 56.4.2. were entirely heuristic.
This is particularly important in the case 56.4.2. where the undesirable result
was obtained in a definite way for a specific instance. Indeed, the treatment
of 56.4.2. referred to the “ preliminary” heuristic discussion of the essential
zero-sum three-person game in 21.-23., and not to its exact theory in 32.

What happened there — in 56.4.2. as well as in 56.4.1. — can be described
in the terminology of the exact theory as follows: the general two-person
game T of 56.4.1. led to a zero-sum three-person game T which coincides
with the simple majority game of three persons. The exact theory of
30.1.1. provided various solutions for this game, which were classified and
analyzed in 33.1. Now the considerations of 56.4.1. and 56.4.2. amounted
to selecting a particular one from among these solutions: the non-discrimina-
tory solution of 33.1.3.

Consequently we must ask ourselves: Was it reasonable to select just
this — the non-discriminatory — solution? Is it not possible that another
one among the solutions — i.e. a discriminatory one in the sense of 33.1.3. —
is free from the objections which hold us up?

66.6. The Discriminatory Solutions

66 . 6 . 1 . If we had approached the essential zero-sum three-person game —
i.e. the simple majority game of three persons — from any other angle, and
if it had been necessary to select a particular one from among its solutions,

1 We made repeated use of such replacements in 56.4.1., but not in the subsequent
argument of 56.4.2. !

512

GENERAL NON-ZERO-SUM GAMES

there would have been a strong presumption in favor of the non-discrimina-
tory one. This solution — i.e. the standard of behavior which it represents
— gives all three players equal possibilities to compete for coalitions, and
in the absence of any definite motive for discrimination one is tempted to
treat it as the most “natural” solution of this game. 1

However, in our present situation there is every reason to discriminate:
In the game T, players 1,2 are real players, the original participants of T,
while player 3 is, as repeatedly emphasized, just a formalistic fiction.
Throughout the discussion of the preceding paragraphs we have stressed
that this player should not compete for a coalition, and that he should not
be treated like the others. In other words, if we expect to be able at all
to apply the theory of 30.1.1. to this situation, then there is an absolute
necessity of discriminating against the fictitious player 3, — i.e. to choose one
of those solutions which were termed discriminatory in 33.1., the excluded
player being the fictitious player 3.

We saw, loc. cit., that these discriminatory solutions are characterized
by the fact that the excluded player — whom the solution, i.e. the standard of
behavior, disqualifies from competing for a coalition — is assigned a fixed
amount c in all imputations of the solution. It appeared in 33.1.2. that this
amount need not be the minimum at which the excluded player can maintain
himself alone, — i.e. not necessarily c = — 1. Actually c could be chosen
from a certain interval: — 1 ^ c <

66.6.2. At this point it may be useful to interrupt the discussion for a
moment in order to comment briefly upon the discriminatory solution which
excludes the fictitious player in the worst possible situation, — i.e. with
c = — 1. According to 33.1.1. this solution consists of precisely those
imputations in which the fictitious player 3 gets — 1, and each one of the two
real players gets ^ — 1.

As pointed out loc. cit. this means that the solution — i.e. the standard
of behavior — restricts in no way the division of the proceeds between the
two real players. The reason given there is now valid in a much more
fundamental way: the bargaining of the players 1,2 has become entirely
unrestricted, not only because the accepted standard of behavior excludes
the interference of player 3 — which was the only normative influence in the
relationship of players 1 ,2 — but also for the still better reason that player 3
does not exist. It is easy to see that this removes the threat that player
1 or 2 will forsake cooperation with the other if his “fair share” is not
conceded by his partner and that instead he will cooperate with player 3 and
obtain a compensation from that source.

56.7. Alternative Possibilities

56 . 7 . 1 . Let us now continue the discussion where it left off at the end of
56 . 6 . 1 .

1 Of course the other solutions are just as good in the rigorous sense of 30.1.1., but the
above statement is nevertheless reasonable prima facie.

EXTENSION OF THE THEORY

513

It may seem questionable whether we should insist upon c = — 1, or
allow the entire variability — 1 g c < £. The first alternative is the more
plausible prima fade. Indeed, c > — 1 means that the real players do not
exploit the fictitious player to the utmost of their possibilities, i.e. that they
do not endeavor to gain as much (as a totality) as feasible. One might view
such a self-denial as a compensation paid to the fictitious player by virtue
of the accepted stable standard of behavior. And since we have decided
to exclude any participation of the fictitious player in the interplay of coali-
tions and compensations, there is some justification in forbidding this.

It must be conceded, however, that this argument is not altogether
cogent. A (positive) compensation paid by the fictitious player is a
qualitatively different thing from one paid to him. The former is a patent
absurdity, since the fictitious player does not exist and will therefore not
pay compensations. The latter, on the other hand, is not absurd at all.
It merely expresses a self-denial in exploiting a possible collective advantage,
and we have had several instances showing that a stable standard of behavior
can require such conduct. 1 It is not a priori evident that such a self-denial
is out of the question in the present situation. 2 To exclude it would mean
that a stable standard of behavior — in the presence of complete information
— necessarily entails attainment of the maximum collective benefit. The
reader who is familiar with the existing sociological literature will know
that the discussion of this point is far from concluded.

We shall nevertheless succeed in settling this question within the frame-
work of our theory by showing that c must be restricted to its minimum value. 3

56 . 7 . 2 . For the moment, however, we must develop both alternatives
concurrently.

To this end we return to the general n- person game r, and the correspond-
ing zero-sum (n + l)-person game T. We are now able to formulate the
relevant concepts rigorously.

(56:A:a) Denote the set of all solutions V of T by Q.

(56:A:b) Given a number c, denote the system of those solutions V

of T for which every imputation a = {ai , • • • , a n , a n+ i) of
V has a n+ 1 = C, 4 by fl c .

1 This is, of course, just another way to express the possibility of 33.1.2. Another
instance, in a zero-sum four-person game, is given in (38 :F) of 38.3.2. Still another
obtains for ail decomposable games in 46.11. (In this last instance the self-denial is
exercised by the players of A when < 0, and by those of H when v > 0, cf. loc. cit.)

We emphasize that such self-denial is exercised under the pressure of the accepted
standard of behavior, although the players are assumed — as always in our theory — to be
informed fully about the possibilities of the game.

1 However if it occurred, it would be regarded normally as an inefficient — though
stable — form of social organization.

* I.e. the self-denial in question does not occur and the maximum social benefit is
always obtained. This result is not as sweeping as it may seem, since we are assuming a
numerical and unrestrictedly transferable utility, as well as complete information.

4 I.e. where the fictitious player gets the same amount c in all imputations of the
solution.

514

GENERAL NON-ZERO-SUM GAMES

(56:A:c) Denote the sum of all sets ft c by ft'.

(56:A:d) Denote the ft c of c = v((n + 1)) = — v((l, * • * , n)) by

ft". 1

In connection with (56 : A :c) we note this :

For some c the set ft c is empty. These c may obviously be omitted
when ft' is formed. Thus a n +i ^ v((n + 1)) = —v((l, • • • , n)) necessi-
tates c — v((l, • • • , n)), otherwise ft c is empty. Again

«.+i = - X ^ - X

k-i k - i

n

^ v((A)), otherwise ft c is empty. So c is subject to

*-i

v((l, ••',n))gci- £ v((fc)).

fc-1

Actually it is usually even more restricted. 2

The ft" of (56:A:d) belongs to the minimum c of (56:4).

58.8. The New Setup

56.8.1. Our discussion of 56.3.2.-56.4.3. showed that the solutions of ft
are certainly not all significant for T. The analysis of 56.6.1. restricted
those solutions further, but it left the question unanswered whether the
system of all significant solutions is ft' or ft".

Thus the systems ft' and ft" correspond to the two alternatives referred
to.

We now proceed to differentiate between ft' and ft".

Consider the imputations

(56:5) a = {a h • • • , a n , a n +i}

of the game T. Among the components a h • • • , a ny a n +i the n first
ones, ai, • • • , a n express realities: the amounts which the real players,
1, • • • , n respectively, are to obtain from this imputation. The last
component, a n +i on the other hand, expresses a fictitious operation: the
amount attributed to the fictitious player n + 1. Further, this component

1 1.e. where the fictitious player gets, in all imputations of the solution, only that
amount which he could obtain for himself even in opposition to all others. This means —
as we know — that the real players obtain together the maximum collective benefit.

* Thus in the essential zero-sum three-person game, (56:4) gives

-1 ^ c £ 2,

while we know from 32.2.2. that the exact domain of c (with non-empty is

-1 £ c < |.

necessitates c ^ —
the restriction
(56:4)

EXTENSION OF THE THEORY

515

a n+ i is not only fictitious in the interpretation of T, but it is also mathe-
matically unnecessary, — i.e. it is determined if the on, • • • , a n are known.

Indeed (since the sum of all components of the imputation a must be
zero)

(56:6) a n +i = — X

1

Consequently it may be preferable to express a by specifying its compon-
ents a i ,•••,<*» only, always remembering that a n +i can be obtained — if
desired — from (56:6). Thus we shall write

(56:7) a = { {on, * * • , a n ) }.

We observe that this notation is not intended to supersede the original one, —
i.e. we wish to be free to use both (56:5) and (56:7), whichever may be more
suitable. Indeed, it is in order to avoid misunderstandings which might
ensue from this double notation, that we are using the double brackets
{ { | } in (56:7) instead of the simple ones { | in (56 -.5). 1

56 . 8 . 2 . The imputation a in its form (56 :5) was subject to the zero-sum
restriction, and also to the restriction

(56:8) a, ^ v((t)) for i = 1 , • • • , n, n + 1 .

We must express (56:8) for (56:7) (with (56:6)).

Now for i = 1, • * • , n (56:8) is unaffected by the transition from
(56:5) to (56:7), but for i = n + 1 we must make use of (56:6). So it
becomes

X «< ^ —v((» + 1)) = v((l, • • • , n)).

«-l

1 Of course we could have done this all along, i.e. for the original zero-sum n-person
games. Here an imputation

a “ |oi, • • • , <* n \

is determined if only its components i ^ i 0 are given (for any fixed i 0 ), since

In conformity with this we have observed already in (31:1) in 31.2.1. that the imputa-
tions of the (essential) zero-sum n-person game form an (n — l)-dimensional, and not an
n-dimensional, manifold.

However, there was no particular advantage to be gained by getting rid of an « J# ,
and there was no way to decide which a* # should be eliminated, if any. In the graphical
discussion of the essential zero-sum three-person game we actually made an effort to
keep all a, in the picture. (Cf. 32.1.2.)

The situation now is altogether different, considering the special role of a*+i. The
elimination of <*„ + 1 will be essential for our subsequent deductions.

516 GENERAL NON-ZERO-SUM GAMES

Thus (56:8) goes over into this:

(56:9) cti ^ v((t')) for * = 1, • • • , n;

(56:10) X ^ • • • > »))•

• -1

56.0. Reconsideration of the Case Where r is a Zero-sum Game

56 . 9 . 1 . Let us stop for a moment to interpret these restrictions.

(56:9) is not new. It expresses again what we had already for the
zero-sum games, namely that nobody would accept in any case less than he
can get for himself in opposition to all others. (56:10), however, appears
for the first time. Its meaning becomes transparent if we consider the
quantity v((l, • • • , n)) more closely.

v((l, • • * , n)) is the value of the game for the composite player
comprising all real players 1, • • • , n, and playing against the fictitious
player n + 1. The amount which this composite player gets at the end
of a play is, of course

5 ) Wi k(r h • • * , r n ).

He controls the variables ti, • • • , r n , i.e. all the variables which occur
in this expression. Thus in the zero-sum two-person game the real players
control all moves the fictitious player having no influence on the course
of the game.

In comparing this with the zero-sum two-person scheme described in

n

14.1.1. our ^ 3C* corresponds to 3C there, all our variables ti, • • • , r n

jfc-i

to the one variable t y there, while no domain of variability in our present
set-up corresponds to the variable r 2 there.

It is intuitively clear that the value of such a game (for the first player)
obtains by maximizing with respect to all variables (since they are all
controlled by him). This is

n

(56:11) Max, t X 3C *( T ‘> ‘ * * . *«)

*-l

in our set up, the corresponding expression in the scheme of 14.1.1. being
(56:12) Max Tj 3C(ri, r 2 ) (r 2 is really absent).

Of course the systematic theory of 14., 17. gives the same result: Vi, v 2
in 14.4.1. are equal to each other and to (56:12), since the operation Min Ti
is void. So the game is strictly determined and has the value (56:12) in

EXTENSION OF THE THEORY 517

the sense of 14.4.2. and 14.5. Consequently the general theory of 17.
yields necessarily the same value.

So we see:

(56:13) v((l, •••,»)) = Max,, £ 3C*( >,, • • • , r n ).

k-1

Consequently (56:10) expresses this: No imputation should offer all (real)
players together more than the totality can expect in the most favorable
case, i.e. assuming complete co-operation and the best possible strategy. 1,1
Summing up:

(56 :B) The imputations of (56:7) are subject to the following

restrictions:

(56:B:a) No real player must be offered less than he can obtain for

himself even in opposition to all other players (cf. (56:9)).
(56:B:b) All real players together must not be offered more than the

totality can expect in the most favorable case, i.e. assuming
complete co-operation and the best possible strategy (cf.
(56:10) and (56:13)).

This formulation makes the common-sense meaning of our restrictions
(56:9), (56:10) (i.e. (56:B:a), (56:B:b)) quite clear: A violation of (56:9)
(i.e. of (56:B:a)) means that one of the (real) players receives an offer which
is more unfavorable than what can be enforced against him. A violation of
(56:10) (i.e. of (56:B:b)) means that the totality of all (real) players receives
an offer which is more favorable than it could ever expect to achieve. It
seems reasonable to consider these as precisely the conditions under which
players who act rationally will refuse to consider a distribution scheme (an
imputation) because it is manifestly absurd.

56 . 9 . 2 . Before proceeding any further we must once more retrace our
steps and compare our present set up with the previous one, in the cases
where both apply.

Specifically: Assume that we are applying the procedure of the past
sections to an n- person game r which is already zero-sum. Accordingly
we form for this game the zero-sum n + 1-person game T as described in
56.2.2., and then proceed as in 56.8.2.

1 Note that the concept of a best strategy for the totality of all real players is clearly
defined: if there is complete co-operation, then the totality faces a pure maximum
problem.

1 If the game in its original form — i.e. before the normalization of 12.1.1. and 11.2.3.
is performed — contains chance moves, then the “most favorable case” referred to above
must not be taken to include these too. I.e. only co-operation and optimal choice of
strategies is to be assumed, while the chance moves must be accounted for by forming
expectation values. Indeed, it is in this way that we passed in 11.2.3. from the

S*(T0, Ti, • • * , T n )

(ro representing the influence of all chance moves) to the 3C*(n, • • • , r*) which we are
using now.

518

GENERAL NON-ZERO-SUM GAMES

It is important not to misunderstand the meaning of this operation.
Obviously the operations of 56.2.2. and 56.8.2. are entirely unnecessary if
T itself is a zero-sum game; we possess a theory which disposes of this case.
But if a more general theory, valid for all games, is to be constructed on
this basis, then we must demand that it agree with the (more special) old
theory as far as the latter goes. I.e. in the domain of the old theory, where
the new theory is superfluous, the new must agree with the old. 1

56.9.3. That T is a zero-sum n- person game means

n

£ 3C *(t,, • • • , r„) = 0,
k- 1

i.e. 3C n +i(ri, • • • , t») s 0. Thus v(£) is not affected if the fictitious
player n + 1 is added to (or removed from) the set S. I.e.:

(56:14) vOS) = vOS u (n + 1)) for S £ (1, • • * , n).

The special cases S = ©, (1, • • • , n) give

(56:15) v((n + 1)) = 0,

(56:16) v((l, •••,» + 1)) « 0.

(56:14), (56:15) together show that the game T is decomposable with
the splitting sets (1, • • • , n) and (n + 1). Its (1, • * * , n) constituent
is the original game T, while the fictitious player n + 1 is a dummy. 2
(For the decomposition cf. the end of 42.5.2. as well as 43.1. For the
dummies cf. footnote 1 on p. 340, and the end of 43.4.2.)

Now we can observe:

56.9.4. First: Since T obtains from T by the addition of a dummy, the
solutions of T and T (in the old theory) correspond to each other, the
only difference being that the latter takes care of the dummy (the fictitious
player n + 1) also, assigning him the amount v((n + 1)), i.e. zero. (Cf.
46.9.1. or (46 :M) in 46.10.4.)

Our proposed new theory would obtain the solutions for T from the
(old theory) solutions of T. Hence the above consideration proves that all
the new solutions to be obtained for T will be among the old ones. Further-
more we see that in this case we can — indeed must — take the entire system
SI of (56:A:a) in 56.7.2. It must be noted, however, that in this case all
solutions of SI automatically assign the fictitious player n + 1 the amount
v((n + 1)). I.e. here SI = Sl c with c = v((n + 1)), i.e. SI = Q". (Cf.

1 This is a well known methodological principle of mathematical generalization.

* The reader should recall that the fictitious player is not, in general, a dummy in the
game f . This may sound paradoxical, but it was established in 56.3. for the very special
case of the general two-person games r. Indeed, it is just because the rules of the game F
do not in general assign him the role of a dummy that we must restrict the solution V
of F to those which do restrict him to such a role. This is the meaning of the discussions
of 56.3.2.-56.6.2.

We shall determine in 57.5.3. which properties of V are necessary and sufficient in
order that the fictitious player be a dummy.

EXTENSION OF THE THEORY

519

(56:A:b) and (56:A:d), loc. cit.) Consequently any sets we might define
between ft and ft" — in particular, both ft' and ft" of (56:A:c) and (56:A:d),
loc. cit. — coincide with ft and are equally acceptable for our purpose.

In other words: the choice between ft' and ft" which is still ahead of us
is of no significance in this case. Both alternatives here are in agreement
with the old theory; indeed, there is no need here to abandon the old theory
at all. 1

56.9.5. Second: The imputations for a zero-sum n- person game were
defined in the old theory in this way:

a = {«i,

a. ^ v((i)) for i = 1, ■ • • , n;

1 a. = 0.

>-i

Our new arrangement of (56 :7) in 56.8. 1 . differs from this. Here we have

(56:C:a*) a — {{a t , ■ • •

and by (56:9), (56:10) and (56:16),

(56:C:b*) ^ v((i)) for i = 1, • • • , n;

n

(50:C:c*)

»-l

We already know from the preceding remark that there can be no real
difference in the present case between the old theory and the new one. 2
It is nevertheless useful to see directly that (from the point of view of the
old theory) the two procedures (56:C:a)-(56:C:c) and (56:C:a*)-(56:C:c*)
contain really no discrepancy.

The only difference between these two arrangements lies in (56:C:c)
and (56:C:c*). Recalling the definitions of 44.7.2., we see that the differ-
ence between (56:C:a)-(56:C:c) and (56:C:a*)-(56:C:c*) can also be stated
in this way: The first amounts to considering solutions for E( 0) ; the second,
to considering solutions for F( 0). Now we have noted in 46.8.1. that 0 lies
in the "normal" zone of the game F, and by (45:0:b) in 45.6.1., E( 0) and
F( 0) have the same solutions. Thus we have a perfect agreement.

These two remarks made systematic use of the theory of composition
and decomposition of Chapter IX, in order to analyze the influence of our
contemplated new procedure on the zero-sum games r. This procedure
consisted mainly in the passage from T to T, which as we saw amounted to
the addition of a dummy to r. This is considerably more special than the
general compositions dealt with loc. cit. The specific results used could

1 The necessity of restricting ft was deduced in 56.5.-66.6. by considering a non-iero
sum game r.

1 Or rather any new one built along the lines contemplated — we have not yet made the
decision between ft' and ft".

(56:C:a)

(56:C:b)

(56 :C :c)

520

GENERAL NON-ZERO-SUM GAMES

accordingly have been obtained with less effort than by the use of the far
more general theorems referred to. We shall not enter into this subject
further since the general results of Chapter IX are available in any case, and
because the above treatment projects our present considerations more
clearly upon their proper background.

66.10. Analysis of the Concept of Domination

56 . 10 . 1 . We now return to the general n- person game F, its zero-
sum extension T, and the new treatment of imputations as introduced
in 56.8.

Certainly all solutions of T in general cannot be used to define a satis-
factory concept of solutions for T. This was established by the consider-
ation of a special case — i.e. by casuistic procedure — in 56.5.-56.6. Let us
now approach this problem systematically; i.e. apply to the game the formal
definition of a solution as given in 30.1.1. and try to determine in full
generality which of its features are unsatisfactory and require modification.

In doing this we shall use the concept of imputation (of T) in the new
arrangement (56:7) in 56.8.1. The important point about this arrangement
is that it stresses ab initio the primary importance of the real players in I\ —
i.e. directs our attention more to T than to I\ This does not impair, of
course, the fact that we apply the formal theory of 30.1.1. to the zero-sum
n + 1-person game T, and not to the general n-person game T (which
would not be possible).

The concepts of 30.1.1. are all based on that of domination. We there-
fore begin by expressing the meaning of domination as defined loc. cit. for
imputations (of T) with the new arrangement of (56:7) in 56.8.1.

Consider two imputations

= U«l,

,)), 0 = {{ 0 „ • • •

Domination

a h P

means that there exists a non-empty set S £ (1, • • • n, n + 1) which is
effective for a , i.e.

(56:17)
such that

2) g v(«s),

* in S

(56:18) a, > ^ for all i in <S.

We wish to express this in terms of the a it pi with i = 1, • • • , n alone.
It is therefore necessary to distinguish between two possibilities:

56 . 10 . 2 . First: S does not contain n + 1. Then

(56:10) S £ (1, • • • * n), S not empty.

EXTENSION OF THE THEORY

521

The conditions (56:17), (56:18) above need not be reformulated since they
involve only the a if ft with i = 1, • • • , n. Besides Sfi(l, •••,»)
in the v(S) of (56:17).

Second: S does contain n + 1. Put T = S — (n + 1). Then

(56:20) Tfi(l, •••,»), T may be empty.

The conditions (56:17), (56:18) above must be reformulated since they
involve a n +i, 0»+i.

It is natural to form — S in (1, • • • , n, n + 1), i.e. as (1, • • • , n,
n + 1) — S; and — T in (1, • • • , n); i.e. as (1, • • • , n) — T. These
two sets are clearly equal, but it is nevertheless useful to have symbols for
both. We denote the first by ±S and the second by — T.

n+ 1

Since a< = 0, so

»-i

5) a. = — = — 5/ a *>

t in S * in _L S i in — T

v(S) = -v(JLS) = -v(-T).

Hence (56:17) becomes

(56:21) £ a,^v(-T).

» in — T

This involves only the a, with i = 1, • • • , n. Besides — T £ (1, • • • , n)
in the v( — T) of (56:21). Next (56:18) becomes

(56:22) a< > ft for all i in T,

and

This last inequality means that

n n

(56:23) £ «< < £ ft.

»-l i-1

(56:22), (56:23) also involve only the a,, ft with t = 1, • • • , n.

Summing up:

— ► — ► %

(56 :D) a H /9 means that there exists either

(56:D:a) an S with (56:19) and (56:17), (56:18);

or

(56:D:b) a T with (56:20) and (56:21), (56:22), (56:23).

Note that these criteria involve only sets S, T, — T £ (1, • • • , n) and the
«i, Pi with t =* 1, • • • , n. I.e. they refer only to the original game T
and to the real players

522

GENERAL NON-ZERO-SUM GAMES

56 . 10 . 3 . The criterion (56 :D) of domination was obtained by a literal
application of the original definition of 30.1.1., the application being made
directly to T and then translated in terms of r. This rigorous operation
having been carried out, let us now examine the result from the point of
view of interpretation; i.e. let us see whether the conditions of (56 :D)
produce a reasonable definition of domination for the present case.

According to (56 :D) domination holds in two cases (56:D:a) and
(56:D:b).

(56:D:a) is merely a restatement of the original definition of 30.1.1. 1
It expresses that there exists a group of (real) players (the set S of (56:19)),

each of whom prefers his individual situation in a to that in P (this is
(56:18)), and who know that they are able as a group, i.e. as an alliance, to
enforce this preference of theirs (this is (56:17)).

(56:D:b) on the other hand is, when viewed in terms of T and of the
real players alone, something entirely new. It requires again that there
exist a group of (real) players (the set T of (56:20)) each of whom prefers

his own individual situation in a to that in p (this is (56:22)). The
ability of this group to enforce the preference in question (i.e. (56:17)) is
not required. Instead we have the condition that the real players left
out of this group must not be able to block the preferred imputation in
question, that is insofar as it affects them (this is (56:21)). 2

Finally there is the peculiar condition that the totality of all (real)
players — i.e. society as a whole — must be worse off under the (preferred)

regime a than under the (rejected) regime P (this is (56:23)).

66 . 10 . 4 . This strange alternative (56:D:b) was, of course, obtained by
treating the fictitious player n + 1 as a real entity. If we refrain from

1 Applied, however, to the general game r, for which that theory was not intended !

* The (real) players left out, i.e. those of —7*, could block the preferred imputation

a if they could get for themselves separately more than a assigns to them together; i.e. if

V a, < v( — T).

(Note that we had to exclude equality here, since that would not block a .) The negation
of this is indeed:

( 66 : 21 )

y a v(— T).

This may be compared with the expression of the ability of the original group T to
enforce its preference, i.e.

(56:17)

y O. s y(T).

It should be noted that neither of (56:17), (50:21) implies the other: It is perfectly
possible that the group T can enforce a as far as it affects the members of T , and that at

EXTENSION OF THE THEORY

523

doing that and try to appraise matters in terms of realities — i.e. of the real
players — then it becomes very difficult to interpret (56:D:b). The best
one can say of it is that it seems to assume the effective operation of an
influence which is definitely set to injure society as a whole (i.e. the totality
of all real players). Specifically in this case domination is asserted when
all players of a certain group (of real players) prefer their individual situa-
tion in a to that in 0 , if the remaining (real) players cannot block this
arrangement, and if it is definitely injurious to society as a whole.

In comparing this domination (56:D:b) to the ordinary one (56:D:a)
the following differences are particularly conspicuous : First, that in (56:D:a)
the ability to enforce one's preference is essential, while in (56:D:b) the
essential point is the ability of the others to block it. Second, that in
(56:D:a) the active group had to be a non-empty set, whereas in (56:D:b)
it could also be an empty set (cf. (56:19) and (56:20)). Third, the anti-
social viewpoint figures in (56:D:b), but not at all in (56:D:a).

The reader will have noticed by now that (56:D:b) is of a rather irra-
tional character, but nevertheless not altogether unfamiliar. It would be
easy to enlarge upon the images and allegories of which (56:D:b) is an exact
formalization. There is no need to dwell further upon this subject here.
What matters is that we have every reason to see in the alternative (56:D:b)
the general cause for those difficulties for which a special case was analyzed
in 56.5.-56.6. Clearly (56:D:b) is not an immediately plausible approach
to the concept of domination in the sense in which (56:D:a) is.

We shall therefore try to resolve our difficulties by the simple expedient
of rejecting (56:D:b) altogether.

56.11. Rigorous Discussion

56 . 11 . 1 . We have decided to redefine domination by rejecting (56:D:b)
and retaining (56:D:a) in (56 :D) of 56.10.2. This new concept of domina-
tion can be stated in two ways which both seem to deserve consideration.

First: As pointed out at the beginning of 56.10.3., (56:D:a) amounts to
a repetition of the corresponding definition of 30.1.1. The only difference

the same time the group — T can block a as far as it affects the members of — T. On the
other hand it is also possible that neither group can enforce or prevent anything.

However, if T is a sero-sum game, and if we require (as in the old theory)
then (56:17) and (56:21) are equivalent. Indeed, in this case

0 ,

v(T) +v(-T) - v((l, • • • , n)) -0,

so

2 - - X “•> v( — T) - -v(T),

* in — T i in T

from which the equivalence follows as asserted.

524 GENERAL NON-ZERO-SUM GAMES

is that then T was a zero-sum n- person game, while now it is a general
n-person game!

Thus our present procedure means that we extend to the present case
the definition of domination in 30.1.1. unchanged, irrespective of the fact
that the game is no longer required to be of zero sum. 1

Second: Let us now view the restriction to (56:D:a) from the stand-
point of T rather than of T. Our original discussion in 56.10. yielded
the two cases (56:D:a) and (56:D:b) depending upon the following disjunc-
tion. In the sense of 30.1.1. the domination in T had to be based on a set S.
Now (56:D:a) obtains when n + 1 does not belong to S f while (56:D:b)
obtains when it does. Hence the restriction to (56:D:a) amounts to requir-
ing that the set S must not contain n + 1.

We repeat: Our new concept of domination means, in terms of V,
that in the definition of domination in 30.1.1. we add to the conditions
(30:4:a)-(30:4:c) imposed upon the set S ) the further condition that S must
not contain a specified element, namely n + 1.

This can also be construed as a restriction on the concept of effectivity
loc. cit. : We regard a set S as effective only if it does not contain n -f 1.
(Of course, the original condition (30:3) loc. cit. is also required.)

56.11.2. We now proceed to study the new concept of solution for T,
i.e. for T, based upon the new concept of domination, introduced in 56.11.1.
In analyzing it we shall rely upon the game T and the form (56:5) of impu-
tations (rather than upon the game r and the form (56:7) of imputations)
and the definition of domination as formulated in the second remark of
56.11.1.

We obtain our result by proving four successive lemmas:

(56 :E) If V is a solution for T in the new sense, then every

— ►

a = {<*i, * * • , a n , a n +i)
of V has a n + 1 = v((n + 1)).

Proof: Assume the opposite. Necessarily a n +i ^ v((n + 1)), hence
there would exist an a ={<*!,*•• , a n , a n fi| in V witha n +i > v((n + 1)).
Put a n +i = v((n + 1)) + 6, € > 0. Define /3 = {0i, • • * , 0 n , 0»+i} with

Pi = cti + - for i = 1, • • • , n;

Tv

0 n +i = a„ +1 - t = v((n + 1)).

1 It may seem peculiar that it took us so long to reach this simple principle, — in fact
we need the further considerations of 56.11.2. before we accept it finally. However, the
act of taking over the definition of 30.1.1. without any alternatives, in spite of the
extremely wide generalization which is now performed, requires most careful attention.
The detailed inductive approach given in these paragraphs seemed to be best suited for
this purpose.

EXTENSION OF THE THEORY

525

Since £ 0 < = -j 8 n +i = -v((n + 1 )) = v((l, •••,»)) and ft > a< for

t-1

t = 1, * • • , n, so the use of S = (1, • • • , n) establishes p H a , l As

a belongs to Vf P cannot belong to it. Hence there exists a 7 in V

with 7 H p . Now consider the set S which enforces this domination.

Since S does not contain n + 1 , we have Ss(l, • • • , n). As pi > a»

— ► — — > — ► — > — > — ► __

for 1 = 1 , • • • , n, y H 0 implies 7 H a . But 7 , a are both in V;

hence we have a contradiction.

(56 :F) If V is a solution for T in the new sense, then it is so also

in the old sense.

Proof : We must show that (30:5 :a), (30:5:b) of 30.1.1., with domination
in the new sense here imply the same with domination in the old sense.
Now domination in the new sense implies domination in the old sense; hence
our assertion concerning (30:5:b) is immediate. So only (30:5:a) requires
closer inspection.

Assume therefore that (30:5:a) is invalid in the old sense, i.e. that for

two a , P in V, a H p in the old sense. Let S be the set which enforces
this domination. By (56 :E) a n +i = /3 »+ 1 (= v((n + 1))), h^nce n + 1

cannot belong to S. Consequently a H P in the new sense, i.e. (30:5:a)
fails in the new sense too. This completes the proof.

(56 :G) If V is a solution for T in the old sense, and if every

— ►

« = { Oil , •*•,««, a n +l}

of V has a n +i = v((n + 1 )), then V is also a solution in the new
sense.

Proof: We must show that (30:5 :a), (30:5:b) of 30.1.1. with domination
in the old sense here imply the same with domination in the new sense.
Now domination in the old sense is implied by domination in the new sense;
hence this time our assertion concerning (30:5:a) is immediate. So only
(30:5:b) requires closer inspection.

Consider therefore an a = {ai, • • • , a n , a„+i} not in V. As (30:5 :b)
holds in the old sense, there exists a P = {0i, • • • , 0„, p n + 1 } in V with

—4 — >

p h a in the old sense. Let S be the set which enforces this domination.

Necessarily a n +i ^ v((n + 1)), and since p belongs to V by assumption,
p n +i = v((n + 1)). Hence p n +i ^ a n +i and so n + 1 cannot belong to S.

Consequently p H a in the new sense, i.e. (30:5:b) holds in the new sense
too. This completes the proof.

1 This domination as well as all others in this proof are in the new sense.

526

GENERAL NON-ZERO-SUM GAMES

(56 :H) S? is a solution of T in the new sense, if and only if it belongs

to the system ft" of (56:A:d) in 56.7.2.

Proof: The forward implication results from (56:E) and (56:F), the
inverse implication from (56 :G).

56 . 11 . 3 . In interpreting the result of (56 :H) we must remember that the
discussion originated from the necessity of restricting the system 12 of all
solutions of T for the purposes of the theory of T. We saw in 56.7. that the
plausible result of this restriction should be the set 12' or 12" (or possibly
some set between the two). Thereafter our effort was directed to making
a decision between these two possibilities. Furthermore we concluded
in 56.10.-56.11.1. that a modification of the concept of domination in T
might answer our problem. Now the statement of (56 :H) is that this
modification of the concept of domination leads precisely to the set 12".
By these concurrent results the decision is clearly indicated. We accept
12" as the system of all solutions for T.

56.12. The New Definition of a Solution

66 . 12 . We reformulate this together with references to the main results
on which the decision was based :

(56:1)

(56:I:a) For a general n-person game T, a solution is any solution

(in the original sense of 30.1.1.) of its zero-sum extension, the
zero-sum n + 1-person game T, for which all

— ►

a = {«i, • • * , a„, a»+i j

in V have

(56:24) flfn+i = v((n + 1)).

These solutions form precisely the set fi" of (56:A:d) in 56.7.2.

(56:I:b) Using the form (56:7), a = { {«i, • '• • , a n ) } for these

imputations (i.e. emphasizing r and its players rather than T),
transforms (56 :24) above into

(56:25) £ - v((l, • • • , n)).

»-l

This is clearly a strengthened form of (56:10) in 56.8.2.

(56:I:c) In the special case in which r is itself a zero-sum game, our

new concept of solutions (for T) coincides with the old one, —
i.e. the unmodified application of 30.1.1. (Cf. the first remark
in 56.9.4.) Thus it is no longer necessary to distinguish
between the old theory and the new one. (Cf. also footnote 1
on p. 518.)

THE CHARACTERISTIC FUNCTION

527

For a general n- person game F, the solutions can also be
obtained by applying the definitions of 30.1.1. (which were
intended for zero-sum games only) to T directly and without
any modification. The concept of imputations for T must then
be used in the form (56:7). (Cf. the first remark in 56.11.1.)

The validity of (56:I:d) means that nothing must be added
to the characterization of the imputations in the form (56:7)
as given in 56.8.2. However, by (56:I:b) the equation (56:25)
will then automatically hold in each solution V. Hence we
may, if we wish, add (56:25), — i.e. strengthen (56:10) of
56.8.2. to (56 :2s). 1

The restriction imposed in (56:I:a) upon the solutions of T
can also be expressed by modifying the concept of domination
for F but then allowing all solutions in the modified sense. This
modification consists of imposing upon effective sets (in the
sense of 30.1.1.) the further requirement that they must not
contain n + 1. (Cf. the second remark of 56.11.1.)

57. The Characteristic Function and Related Topics

67.1. The Characteristic Function : The Extended and the Restricted Forms

67.1. We are now in the possession of a theory which applies to all games
and — like the theory of 30.1.1. for zero-sum games, of which it is an exten-
sion — is based exclusively upon the characteristic function. I.e. the
functions 3C*(ri, • • • , r„), k = 1, • • • , n of 11.2.3., which actually define
the game, do not affect the theory directly, but only through the character-
istic function v(S). 2

There is, however, a difference between the use of the characteristic
function v(S) for a zero-sum game, and for a general game. For a zero-sum
n- person game T the characteristic function v(/S) is defined for all sets
Sz (1, • • • , n) and for only these. (Cf. 25.1.) For a general n- person
game T we had to form its zero-sum extension, the zero-sum (n + l)-person
game T, and the characteristic function v(S) was actually formed like the
characteristic function (in the old sense) of I\ (This is the v(S) which

1 This permissibility of restricting

n

X «« S v((l, • • • , »))
n

X «> - v (( i > •••.»))

i-1

is analogous to (but more general than) the equivalence of E( 0) and F( 0) referred to in
the second remark of 56.9.5.

1 Of course v(S) is defined with the help of the 3C*(ri, • • • , r n ). Cf. 25.1.3. and 58.1.

(56:10)

to

(56.25)

(56:I:d)

(56:I:e)

(56:1 :f)

528

GENERAL NON-ZERO-SUM GAMES

figured in all of our recent discussions, particularly throughout 56.4. L,
56.5.1., 56.7.2., 56.8.2., 56.9.1., 56.9.3.-56.10.3., 56.11.2.-56.12.) Accordingly
v(£) is now defined for all sets /Sfi(l, • * • , n, n + 1) and for only these.
We may, however, if we wish, consider v(S) for the sets S £ (1, • • • , n)
only. When this is done, we shall speak of the restricted characteristic func-
tion; while v(S) in its original domain, embracing all S £ (1, • • • , n, n + 1)
is the extended characteristic function.

From this we conclude in the special case of a zero-sum game: Here
the characteristic function of the old theory is the restricted characteristic
function of the new one. 1

Returning to the general games, we see that the characteristic function
is the basis of our entire present theory. Among the equivalent formula-
tions of that theory, (56 :I:a) in 56.12. uses the extended characteristic func-
tion, while (56:I:d) uses the restricted one.

Consequently our next objective is necessarily the determination of
the nature of these characteristic functions, and of their relationship to each
other.

67.2. Fundamental Properties

67 . 2 . 1 . Consider a general n- person game T, and its two characteristic
functions, as defined above: The restricted one, v(S) defined for all subsets
S of 7 = (1, * • * , n) and the extended one, v(S) defined for all subsets S of
I - (1, • • • , n, n + l). 2

In what follows we must distinguish between two possibilities in our
notations for —S, as in the second remark in 56.10.2. For S&I =
(1, • • • , n, n + 1) we can form — S in 7 , i.e. as 7 — S, while for Ss / =
(1, • • • , n) we can also form — S in 7, i.e. as 7 — S* We denote again
the first set by ±S and the second one by — S.

We propose to determine the essential properties of both characteristic
functions of the general n-person game — just as we did in 25.3. and 26., for
the characteristic function of the zero-sum n-person game.

Consider the extended characteristic function first. Since it is the
characteristic function in the old sense for the zero-sum (n + l)-person
game T, it must have the properties (25:3:a)-(25:3:c) formulated in 25.3.1. —
only with 7 = (1, • • • , n + 1) in place of the 7 = (1, • * • , n) there.
In this way we obtain:

(57:1 :a) v(©) =0,

1 All these distinctions and definitions cannot and do not affect the rigorously estab-
lished fact that for all zero-sum games the two theories are equivalent to each other.
(Cf. (56:I:c) in 56.12.)

* We denote them by the same letter, v, since they have the same value wherever both
are defined.

1 Formed for the same S (of course S £ 7), these two sets are clearly different. Loc.
cit. we claimed that they are equal, but there we formed them for two different sets
8 and T.

(57:1 :b)
(57:1 :c)

THE CHARACTERISTIC FUNCTION

529

v(-LS) = -v(S),

v(S uT) ^ v(fl) + v(D if S n T = ©.

OS, TfiJ).

Consider next the restricted characteristic function. We obtain con-
ditions for it from (57:l:a)-(57:l:c), by restricting ourselves to subsets of J.
This is immediately feasible for (57:1 :a), (57:1 :c), but it is impossible for
(57:1 ib). 1 In this way we obtain:

(57 :2:a) v(©) = 0,

(57:2:c) v(S uT) ^ v(S) + v(5P) if S n T = ©.

OS, r*/)

Note that we cannot replace (57:l:b) by something equivalent for — S.
Indeed, all we can do with — S, is to put T = — S in (57:1 :c). This gives

(57:2:b) v(-S) g v(7) - v(S).

Even if v(/) = 0, which need not be the case, (57:2:b) becomes merely
(57:2:b*) vf-S) ^ -v(fl),

but not the equivalent of (25:3:b) in 25.3.1.

v(-S) - -v(S).

(57:l:a)-(57:l:c) as well as (57 :2 :a), (57:2:c) are, by virtue of their
derivation, only necessary properties of the (extended or restricted) char-
acteristic functions. We must now see whether they are sufficient as well.

57.2.2. If T were an arbitrary zero-sum (n + l)-person game, then we
could conclude from the result of 26.2. that any v(S) which fulfills (57:l:a)-
(57:l:c) is the (old sense) characteristic function of a suitable T, — i.e. the
extended characteristic function of a suitable general n- person game T.
In other words: This would prove that the conditions (57:l:a)-(57:l:c) are
necessary and sufficient — that they contain a complete mathematical
characterization of the characteristic functions of all possible general
n- person games I\

However, T is not at all arbitrary. As we saw in 56.2.2., the (fictitious)
player n + 1 has no influence on the course of the game — i.e. he has no

personal moves; the 3 C*(ti, • • • , r n , r n+ i) do not really depend on his

variable T n + 1 . Furthermore, it is clear from 56.2.2. that this is the only
restriction to which T must be subjected: If in a zero-sum n + 1-person
game T the player n + 1 has no influence on the course of the game, then
we can view T as the zero-sum extension of a general n- person game T
played by the remaining players 1, • • • , n. 2

1 S, ±S cannot be both SI « (1, • • • , n) since one of them must necessarily

contain n + 1.

* I.e. we can treat n + 1 as if he were a fictitious player — as far^as the rules of the
game are concerned. We know, of course, that there are solutions V for f which bring
out the fact that he is a real player (those in 0 but not in Q", cf. (56:A:a)-(56:A:d) in
56.7.2. and (56:I:a) in 56.12.; recall also 56.3.2., 56.3.4.).

530

GENERAL NON -ZERO-SUM GAMES

Consequently the following question arises: (57:l:a)-(57:l:c) are neces-
sary and sufficient conditions for the characteristic functions in the old
sense of all zero-sum n + 1-person games. How must they be strengthened
so as to do the same thing for the (old sense) characteristic functions of all
those zero-sum n + 1-person games in which the player n + 1 has no
influence on the course of the game?

Answering this question would amount to giving a complete mathe-
matical characterization for the extended characteristic functions of all
general n + 1-person games. But then the problem of doing the same
for the restricted characteristic functions would still remain.

It will be seen that by attacking the last problem first, a somewhat
more advantageous arrangement obtains: the first problem can be solved
in a few lines with the help of the latter one. However, our approach will
be dominated by the above considerations.

67.3. Determination of All Characteristic Functions

57 . 3 . 1 . We proceed to prove that the necessary conditions (57:2:a),
(57:2:c) are also sufficient: For any numerical set function v(S) which ful-
fills (57:2:a), (57:2:c) there exists a general n-person game V of which this
v(S) is the restricted characteristic function. 1

In order to avoid confusion it is better to denote the given numerical
set-function which fulfills (57:2:a>, (57:2:c) by v 0 (£). With its help we
shall define a certain general n- person game T, and denote the restricted
characteristic function of this T by v(S). It will then be necessary to prove
that v(S) = vo(iS).

Let therefore a numerical set-function v 0 (aS) which fulfills (57:2:a),
(57:2:c) be given. We define the general n- person game V as follows: 2

Each player k = 1, • • • , n will, by a personal move, choose a subset
Sk of I which contains k. Each one makes his choice independently of the
choice of the other players.

After this the payments are made as follows :

Any set S of players for which

(57 : 3 ) Sk = S for every k belonging to S

is called a ring. Any two rings with a common element are identical. In
other words: The totality of all rings (which actually have formed in a play)
is a system of pairwise disjunct subsets of 7.

Each player who is contained in none of the rings thus defined forms
by himself a (one-element) set which is called a solo set . Thus the totality
of all rings and solo sets (which actually have formed in a play) is a decompo-
sition of 7, i.e. a system of pairwise disjunct subsets of 7 with the sum 7.
Denote these sets by Ci, • • • , C p and the respective numbers of their
elements by n h • • • , n P .

1 The construction which follows has much in common with that of 26.1

* The reader should now compare the details with those in 26.1.2.

THE CHARACTERISTIC FUNCTION 531

Now consider a player k. He belongs to precisely one of these sets
Ci, • • • , C p , say to C q . Then the player k gets the amount

(57:4) 1 Vo (C f ).

n q

This completes the description of the game I\ r is clearly a general
n- person game, and it is clear what its zero-sum extension T is. We empha-
size in particular that in T the fictitious player n + 1 gets the amount

(57:5) - £ vo(C f ).*

0-1

We are now going to show that r has the desired restricted characteristic
function v 0 (/S).

57.3.2. Denote the restricted characteristic function of T by v(S).
Remember that (57:2:a), (57 :2:c) hold for v(S) because it is a restricted
characteristic function, and for v 0 (S) by hypothesis.

If S is empty, then v(S) = v 0 (S) by (57:2:a). So we may assume that
S is not empty. In this case a coalition of all players belonging to S can
govern the choices of its Sk so as with certainty to make S a ring. It
suffices for every k in S to choose his S k = S. Whatever the other players
(in — S) do, S will thus be one of the sets (rings or solo sets) C\, • • • , C p
say C q . Each k in C q = S gets the amount (57 :4), hence the entire coalition
gets the amount v 0 (S). Consequently

(57:6) v(S) ^ Vo (S).

Now consider the complement — S. A coalition of all players k belong-
ing to — S can govern the choices of its k so as with certainty to make S
a sum of rings and solo sets. If —S is empty, then this is automatically
true, since then S = I. If — S is not empty, then it suffices for every
A; in —S to choose his St = —S. Hence —S is a ring, and therefore S
is a sum of rings and solo sets.

Thus S is the sum of some among the sets Ci, • * * , C p say of

cv, • • • , C T '

(1 ',***, r' are some among the numbers 1, • • • , p). Each k in C q
(q = 8 f = 1', • • • , r') gets the amount (57:4), hence the n q players in C q
together get the amount v 0 (C*), and so all players of S together get the

r

amount £ v 0 (CV). Since the CV, • • • , CV are pairwise disjunct sets

s-l

1 The n q players in C 9 get together the amount vo(C fl ) by (57:4); hence all players

p

1, • • , n, i.e. all players in Ci, • • • , C p , get together the amount ^ vo{C g ). Now

0-1

(57:5) ensues.

532

GENERAL NON-ZERO-SUM GAMES

with the sum S, repeated application of (57:2:c) gives vo(C a >) ^ v 0 (£).

«-i

I.e.: Whatever the players of S do, together they get an amount ^ v 0 (S).
Consequently

(57:7) v(S) £ Vo (8).

Now (57:6), (57:7) together give
(57:8) v(S) = vo (5),

as desired.

57 . 3 . 3 . Let us now consider the extended characteristic functions.
Here we know that the conditions (57:l:a)-(57:l:c) are necessary. We
shall prove that they are also sufficient: That for any numerical set function
v(S) which fulfills (57:l:a)-(57:l:c) there exists a general n- person game T
of which this v(S) is the extended characteristic function.

In order to avoid confusion, it is again better to denote the given numer-
ical set function which fulfills (57:l:a)-(57:l:c) by Vo(S). The extended
characteristic function of the general n- person game T which we shall
use will be denoted by v(S).

Let therefore a numerical set function v 0 (aS) which fulfills (57:l:a)-
(57:1 :c) be given. Consider it for a moment f or the sets £ si = (1, ■ • • , n)
only, then it fulfills (57:2:a), (57:2:c). Hence our construction of 57.3.1.,
57.3.2. can be applied to this v 0 (S). So a general n-person game T obtains,
such that its restricted characteristic function has always v(S ) = v 0 (S) 1
and so its extended characteristic function has v(S ) = v 0 (S) for S s /.
I.e., if we revert to the natural domain of these S, 2 then we have:

(57:9) v(S) = v 0 (S) if n + 1 is not in S.

Now let n + 1 be in S. Then it is not in _L>S. Hence (57:9) gives
v( JL S) = vo( ±S). (57:1 :a)-(57 :1 :c) hold for v(S ) because it is an extended

characteristic function, and for v 0 (S) by hypothesis. Therefore (57:1 :b)
gives v(±S) = — v(S), v 0 (J _£) = — v 0 (/S). All these equations combine
to

(57:10) v(S) = voOS) if n + 1 is in S.

Now (57:9), (57:10) together give
(57:11) v(S) = vo OS)

unrestrictedly, as desired.

67 . 3 . 4 . To sum up: We have obtained complete mathematical character-
izations of both the restricted and the extended characteristic functions
v(S) of all possible general n-person games T. The former are described
by (57:2:a), (57:2:c), and the latter by (57:l:a)-(57:l:c).

1 The “always” in this case refers, of course, only to the SSI.

’ Which in this case consists of all SSL

THE CHARACTERISTIC FUNCTION

533

We follow therefore the comparable procedure of 26.2., and call the
functions which satisfy these conditions restricted characteristic functions or
extended characteristic functions , respectively — even when thev are viewed
in themselves, without reference to any game.

67.4. Removable Seta of Players

67.4.1. The result which we obtained for extended characteristic func-
tions can also be stated as follows : Every characteristic function (in the old
sense) of any zero-sum (n + l)-person game is also the extended character-
istic function of a suitable general n- person game. 1 Remembering the
discussion of 57.2.2., this means: Every characteristic function of any
zero-sum n + I-person game is also the characteristic function of a
suitable zero-sum n + l-person game in which the player n + 1 has no
influence on the course of the play.

Let us replace in this statement n + 1 by n, obtaining the equivalent
for zero-sum n-person games and the role of the player n. In order to
formulate this result, it is convenient to define:

(57: A) Let a zero-sum n-person game T and asetSs 7 = (1, • • • , n)

be given. Then we call S removable for T, if it is possible to
find another zero-sum n-person game T', which has the same
characteristic function as T but in which no player belonging to S
has an influence upon the course of the game.

Using this definition, our assertion becomes that the set S = (n) is remov-
able. Given any player A; = 1, • • • , n, we can interchange the roles of the
players k and n, hence the set S = (fc) is also removable. So we see:

(57 :B) Every one-element set S is removable in every game T.

Now it should be noted that according to our theory the entire strategy
of coalitions and compensations in a game depends only on its characteristic
function. Consequently the two games T and T' of (57 :A) are exactly
alike from that point of view.

Hence (57 :B) can be interpreted as follows: The role of any one player in
any zero-sum n-person game — insofar as the strategic possibilities of coali-
tions and compensations are concerned — can be duplicated exactly in an
arrangement which deprives him of all direct influence upon the course of
the game. Here we mean his “role” in the most extended sense: including
his relationship to all other players, and his influence on their relationships
to each other.

In other words: We described in 56.3.2.-56.3.4. a mechanism by which
a player who has no direct influence upon the course of the game can
nevertheless influence the negotiations for coalitions and compensations.
We have now shown in (57 :B), that this mechanism is perfectly adequate

indeed the conditions (57:l:a)-(57:l:e) coincide with (25:3:a)-(25:3:c) of 25.3.1.
with 7 ■■ (1, • • • , n, » + 1) in place of its / — (1, • • • , n).

534

GENERAL NON-ZERO-SUM GAMES

to describe the influence which any player in any game could have in this
respect. This statement must be taken absolutely literally: Our result
guarantees that all conceivable details and nuances will be reproduced.

57 . 4 . 2 . By (57 :B) every player k = 1, •• • , n is removable individually
— i.e. the one-element set S = ( k ) is — but this does not mean that all these
players are removable simultaneously — i.e. that the set

S = J - (1, - • • , n)

is. Indeed we have :

(57 :C) The set S = / is removable if and only if the game r is

inessential.

Proof : That no player k = 1, • • • , n has an influence on the course
of the game r', means that all functions 3 C[(ti, • • • , r n ) are independent
of all their variables r i, • • • , r n — i.e. that they are constants

(57:12) 3C£(ri, • • • , r n ) = <x k .

From this

(57:13) v(S) = £ «* for all ScJ,

kin S

Conversely, if (57:13) is required, it can be secured by (57:12).

Hence (57:13) is the characteristic function of a game T for which such
a T* exists — and (57:13) is precisely the definition of inessentiality.

For n = 1,2 every game T is inessential, hence there the set S = I —
and with it every set — is removable. 1 For n ^ 3 there exist essential
games, and therefore S = I is in general not removable.

Therefore this question arises:

(57 :D) Which are the removable sets for an essential game r?

(57 :B), (57 :C) contain a partial answer : The one-element sets are removable,
the n-element set ( S = I) is not. Where is the dividing line?

57 . 4 . 3 . The upper extreme is reached when all (n — l)-element sets —
and with them all sets except / — are removable. We call such a game
extreme . It is worth while to visualize what this property entails: The
strategic situation in such a game is equivalent to that where only one
player has an influence on the course of the game, and the role of all others
consists merely in trying to influence his decisions. The means of influenc-
ing him is, of course, offering him compensations; the motive is to induce

1 The main result concerning zero-sum two-person games, according to which each
game of this type has a definite value for each player (say v, — v cf. the discussion of
17.8., 17.9.), means just this: It states that the game is equivalent to the fixed payments
v, — v to the two players — and this is an arrangement where neither of them can influence
anything.

In every essential game, on the other hand, there exists the interplay of negotiations
for coalitions and co mpensations — and this excludes the simultaneous removability of
all players.

THE CHARACTERISTIC FUNCTION 535

him to make decisions which are favorable to the player or players who make
the offer.

Now we can prove:

(57 :E) For n = 3: The essential zero-surn three-person game is

extreme.

(57 :F) For n = 4: There exist extreme as well as non-extreme essen-

tial zero-sum four-person games.

More in detail :

(57 :E*) For the essential zero-sum three-person game, all two-

element sets are removable.

(57 :F*) For an essential zero-sum four-person game all three-

element sets are removable, or all but one. 1,2

The proofs of these statements present no serious difficulties, but we do not
propose to give them here.

The results (57 :B), (57 :C), (57 :E), (57 :F) show that a general theory
of removable sets and extreme games is not likely to be very simple. It
will be considered systematically in a subsequent publication.

57.6. Strategic Equivalence. Zero-sum and Constant-sum Games

67.6.1. We have exhausted the usefulness of the zero-sum extension T
of the general n- person game T, and therefore from now on we shall discuss
the theory of general n-person games without referring to that concept.
Consequently, hereafter we shall use only the game r itself and its restricted
characteristic function, — unless explicitly stated to the contrary. For this
reason the qualification “restricted” will be dropped, and we shall speak
simply of the characteristic function of T. This is also in harmony with
our preceding terminology for zero-sum n- person games, since now the old
and the new use of the concept of a characteristic function are concordant.
(Cf. the remarks next to the end of 57.1.)

Considering these arrangements, the definition of the concept of a
solution must be that described in (56:I:d) of 56.12. Imputations are best
defined as described in (56:I:b) and in the last part of (56:I:e) id. It
seems worth while to restate this latter definition explicitly:

An imputation is a vector

(57:14) 7 = {{«!,

the components ai, • • • , a» being subject to the conditions

(57:15) a% ^ v ( (t ) ) for i - 1, * • • , n;

1 Every two-element set is a subset of two three-element sets (remember that n — 4),
and by the above at least one of these is removable. • Hence every two-element set is
removable in any event.

* The parts of the cube Q of 34.2.2. which correspond to these various alternatives
can be explicitly determined.

536

GENERAL NON-ZERO-SUM GAMES

(57:16) £ «i = v(/).‘

1-1

We can now extend the concept of strategic equivalence to the present
setup. This will be done exactly as in 42.2. and 42.3.1., i.e. in analogy to

27.1.1. :

Given a general n- person game r with the functions 3Ck(ri, • • • , r n )
and a set of constants a}, • • • , a® we define a new game T' with the func-
tions 3 C£(ti, • * • , r n ) by

(57:17) 3C£(r i, • • • , r n ) =3C*(ri, • • • , r n ) + a° k .

From this we conclude, exactly as before, that the characteristic functions
v(S) and v'OS) of these two games are connected by

(57:18) v'OS) = v(S) + £ «* # .

k in S

We call two such games, as well as their characteristic functions, stra-
tegically equivalent.

Since we are free of all zero-sum restrictions, the constants a®, • • • , a®
are unrestricted, just as in (42 :B) in 42.2.2.

We note that this strategic equivalence induces an isomorphism of the
imputations of T and T' exactly as in the two previous instances referred to
above. Specifically the considerations and conclusions of 31.3.3. and of

42.4.2. carry over to the present case unchanged, so that it seems unneces-
sary to reformulate them explicitly.

57.5.2. The domain of all characteristic functions (of all general n-person
games) was characterized by the conditions (57:2:a), (57:2:c), which we
restate:

(57 :2 :a) v(©) = 0,

(57:2:c) v(S if T) ^ v(S) + v(T) for S n T = ©.

Among these the characteristic functions of zero-sum games and of constant-
sum games form two special classes. The former are characterized by
(25:3:a)-(25:3:c) of 25.3.1. (Cf. 26.2.) I.e. we must add to our (57:2:a),
(57:2:c) (which coincide with the (25:3:a), (25:3:e) mentioned) the further
condition

(57:19) v(-S) = — v(jS).

The latter are characterized by (42:6:a)-(42:6:c) in 42.3.2. (cf. id.). I.e. we
must add to our (57 :2:a), (57 :2:c) (which coincide with the (42:6:a), (42:6:c)

1 As was pointed out loc. cit. we could have used equivalently

n

^ Oi i V(/).

»-l

Indeed this is the original form of this condition. However, we prefer (57:16).

THE CHARACTERISTIC FUNCTION

637

mentioned) the further condition

(57:20) vOS) + v(-S) - v(I).

Since the zero-sum games are a special case of the constant-sum games,
(67 :20) must be a consequence of (57 :19), always assuming (57:2:a), (57:2:c).
This is indeed so; we can actually prove somewhat more, namely:

(57 :G) (57:19) is equivalent to the conjunction of (57:20) with

v(/) = 0.

Proof: 1 Assuming v(7) = 0, (57:19) and (57:20) are clearly the same
assertion. Hence it suffices to show that (57 :19) implies v(7) = 0. Indeed,
(57:2:a), (57:19) give v (7) = v(-©) = -v(©) = 0.

Note that (57:20) is the assertion that equality holds in (57:2:c) when
S u T = 7. 2 Thus the v(S) of constant-sum games are characterized
by the property that the merger of two distinct coalitions S and T produces
no further profit if together they contain all players.

For the v(S) of zero-sum games the further requirement v(7) = 0
must be added.

To conclude, we emphasize that the extra conditions (57:19) or (57:20)
do not mean that any game with such a characteristic function is neces-
sarily a zero-sum or a constant-sum game. They imply only that such a
characteristic function must belong — among others — to at least one zero-
sum or constant-sum game. It can happen that a game without being
zero-sum (or constant-sum) itself has such a characteristic function, i.e.
the characteristic function of a zero-sum (or constant-sum) game. In
this case it will behave from the point of view of the strategy of coalitions,
and the compensations like a zero-sum (or constant-sum) game without
actually being one.

57.6.3. We are now in the position to settle a question which was in
the foreground several times in our discussions. The analysis of 56.3.2.-
56.4.3. was concerned already with the fact that the fictitious player — in
spite of his unreality — is not ipso facto a dummy. I.e. not one in the sense
of the extended characteristic function and the decomposition theory of the
zero-sum extension I\ 8 This subject came up again at the beginning of
56.9.3., where we noted that he is a dummy for zero-sum games Y.

The question which we will answer now is accordingly this: For which
general games r is the fictitious player a dummy? 4 We prove:

(57 :H) The fictitious player is a dummy if and only if Y has the

same characteristic function as a constant-sum game — i.e. if
(57:20) is fulfilled.

1 Essentially this argument was made in 42.3.2.

‘Indeed S U T — 7 and the usual hypothesis of (57:2:c) S n T * © mean that
T -

* We had to exclude him from the game by explicitly restricting the solutions from
O to fl".

4 The argument of the first remark in 56.9.4., shows then that for these games 0 and
U" coincide — i.e. the restriction of the solutions of f is unnecessary.

538

GENERAL NON-ZERO-SUM GAMES

Proof: As observed at the end of 43.4.2. a player is a dummy, if and
only if he forms (as a one-element set) a constituent of the game. We
must apply this to the fictitious player n + 1 in the zero-sum game I\
That (n + 1) is a constituent, means obviously that

(57:21) v(S) + v((n + l))-v(Su(n+l)) for all Sfi(l, •• •,»).

Now we have

v((n + 1)) = v(7),

v(S u (n + 1)) = — v( J_£ u (n + 1)) = — v( — S).
Hence (57:21) becomes

v(S> - v(7) = -v(-fl),
i.e.

(57:22) v(S) + v(-S) = v(/).

And this is precisely the condition (57:20).

58. Interpretation of the Characteristic Function
58.1. Analysis of the Definition

68.1. We have arrived at a formulation of the theory of the general
n-person game, and found that the concept of the characteristic function
is just as fundamental in it as it was in the preceding theory of the zero-sum
n-person game. It is therefore appropriate to survey the meaning of this
concept once more, putting its mathematical definition into an explicit
form and adding some interpretative remarks.

Consider accordingly a general n-person game T, described by the func-
tions 3 C*(ti, • • • , r n ) (k = 1, • • • , n) in the sense of 11.2.3. The value
v(<S) of the characteristic function for a set S £ / = (1, • • • , n) obtains
by forming this quantity for the zero-sum n + 1-person game T — the zero-
sum extension of T. 1 Hence we can express it by means of the definitory
formulae of 25.1.3.:

(58:1) v(S) = Max- Min- K(7, 7) * Min- Max- K(7, 7),

in n i

where we have:

{ is a vector with the components

Sr* * o, X Sr* = i;

T*

1 We restrict ourselves to the S E / — (1, • • • , n) i.e. to the restricted characteristic
function. The use of all 8 £ / - (1, * • • , n + 1), i.e. of the extended characteristic
function, is contrary to our present standpoint. (Cf. the beginning of 57.5.1.)

INTERPRETATION

539

r\ is a vector with the components tj t -j

Vt " 8 SSs ^ Vt ~ 8 “ I J

r a is the aggregate of the variables r*, k in <S; r _s is the aggregate of the
variables r*, k in — S ; 1 and finally

(58:2)

where

K( { , v ) = X x ( tS> t s )^' 4 »

r*,r " a

(58:3) 3C(t s , t s ) = £ 3 C*(t,, • • • , r„) . s

kin 3

68.2. The Desire to Make a Gain vs. That to Inflict a Loss

68.2.1. K( £ , rj ) is obviously the expectation value of a play of the game

T for the coalition S, if the coalition S uses the mixed strategy £ and the

— ►

opposing coalition — S 8 uses the mixed strategy rj . Hence (58:1) defines
v(S), the value of a play for the coalition S under the assumption that the

coalition S wants to maximize the expectation value K( £ , 17 ), while the
opposing coalition — S wants to minimize it, — and they choose their respec-
— ► — >

tive (mixed) strategies £ , rj accordingly.

Now this principle is certainly correct in the zero-sum n + 1 -person
game T , 4 but we are really dealing with the general n- person game T —

1 — S denotes I — S. Since we are dealing with f , we should have formed _L£ which
is / — S. (Cf. the beginning of 57.2.1.) However, this is immaterial, because no
variable t w+ 1 exists. (Cf. the end of 56.2.2.)

* We use only the original 3C*, k = 1, • • , n, i.e. the 3C n+ i of (56:2) in 56.2.2.

(58:4) 3Cn r l(Tl, • * * , Tn) = ^ «JC*(ti, ' • • , Tn)

k~l

does not occur here. This is, of course, due to the fact that S £ / = (1, • • • , n).

It must be remembered that formula (58:3) above is the first formula of (25:2) in
25.1.3. The second formula of (25:2) loc. cit. gives

(58:5) X (t s ,t~ s ) m - £ X *(r,, • • , T n ).

k in _LS

(Note that we must now definitely use « / — S for the — S loc. cit., since we are
dealing with T. Cf. also footnote 1 above.) Since n 1 is not in S , it is in X$; hence

the sum ^ of (58:5) does contain the3C M i of (58:4), However, (58:4) guarantees,

Ac iti

as it must, the identity of the right-hand sides of (58:3) and (58:5).

3 The observations of footnote 1 above apply again.

4 1.e. if we view —S -l — S as really representing ±S » / — S.

540

GENERAL NON-ZERO-SUM GAMES

T is merely a “working hypothesis”! And in r the desire of the coalition
— S to harm its opponent, the coalition S, is by no means obvious. Indeed,
the natural wish of the coalition — S should be not so much to decrease

the expectation value K( { , r? ) of the coalition S as to increase its own

expectation value K'( £ ,

These two principles would be identical

if every decrease of K( £ , rj ) were equivalent to an increase of K'( £,*?).
This is of course the case when T is a zero-sum game, 1 but it need not
at all be so for a general game r.

I.e. in a general game r the advantage of one group of players need
not be synonymous with the disadvantage of the others. In such a game
moves — or rather changes in strategy — may exist which are advantageous
to both groups. In other words, there may exist an opportunity for
genuine increases of productivity, simultaneously in all sectors of society.

58.2.2. Indeed, this is more than a mere possibility — the situations to
which it refers constitute one of the major subjects with which economic
and social theory must deal. Hence the question arises: Does our approach
not disregard this aspect altogether? Did we not lose this cooperative
side of social relationships because of the great emphasis which we placed
on their opposite, antagonistic, side?

We think that this is not so. It is difficult to present a complete case,
since the validity of a theory is ultima analysi only established by success
in the applications — and we have made no applications in our discussion
thus far. We will suggest therefore only the main points which seem to

1 This is so because when V is zero-sum then

(58:6) K(7, 7) 4* K'(7, T) - 0.

This is clear by common sense; a formal proof obtains in this way: Clearly

k '( 7 , 7 ) - £ x.'(t s , T-s)i T <n T - sl

r^,r -s

3C'(t« t-s) = £ 3C 4 (t„ • • • , t„).

ifcin — 8

(Note that this is not the ^ 3C *(n, • • • , r„) which occurs in (58:5)). Now com-

k in ±8

parison of (58:2) with (58:7) shows that (58:6) is equivalent to
(58:9) 0C(r 5 , r~ s ) + 3C'(r s , r~ 3 ) m 0,

and (58:3), (58:5) imply that (58:9) amounts to

n

X 3C*(n, • • • , Tn) * 0,

*- 1

(58:7)

where

(58:8)

i.e. the zero-sum condition for T.

INTERPRETATION 541

*

support our procedure, and then refer to the applications which provide a
definite corroboration.

58.3. Discussion

58 . 3 . 1 . The following considerations deserve particular attention in
this connection:

First: Inflicting losses on the adversary may not be directly profitable
in a general (i.e. not necessarily zero-sum) game, but it is the way to exert
pressure on him. He may be induced by such threats to pay a compen-
sation, to adjust his strategy in a desired way, etc. Hence it is not a limine
unreasonable that this category of strategic possibilities should be taken
into account; and our procedure in forming the characteristic function, as
analyzed above, might be the proper one to do just that. It must be
admitted, however, that this is not a justification of our procedure — it
merely prepares the ground for the real justification which consists of success
in applications.

Second: A further consideration pointing in the same direction is this.
We have seen that in our theory all solutions correspond to attainment
of the maximum collective profit by the totality of all players. 1 When
this maximum is reached, any further gain of one group of players must be
compensated by an at least equal loss of the others. True, there could be
overcompensation : i.e. one group might obtain a gain by inflicting a greater
loss on the others. However, we have assumed complete information for
all players, and a perfect interplay of threats, counterthreats and compensa-
tions among them. 2 Hence one may assume that such possibilities will be
effective only as threats, and that the corresponding actions will be obviated
always by negotiations and compensations. By this we do not mean that
these threats are “bluffs” which are never “called.” Since there exists
complete information for all players, there can never be any doubt. But
when an action is threatened by which one party gains less than the other
one loses, then there exists ipso facto the possibility of avoiding it by com-
pensations in a way which is advantageous to both sides. 8 And when this
happens it is again true that one side gains exactly what the other loses.

If this argument is accepted as generally valid, then our difficulties
disappear.

58 . 3 . 2 . Third: It may be said that the argumentation of the two pre-
ceding remarks is too sketchy and that it does not justify our theory in the
exact form in which we propose to use it. This is true, but our very detailed

1 Cf. the end of 56.7.1., particularly footnote 3 on p. 513.

* Our entire attitude towards coalitions and compensations was based on this, already
in the theory of the zero-sum games.

1 We do not propose to determine here the amount of the compensation — i.e. the
nature of the compromise. This is the task of the exact theory which we possess already.
It will be the main subject in each application. (Cf. the various interpretations in
61.-63.) At this point we want only to show that actions which would lead to a loss for
the totality of all players, can be avoided by the mechanism described above.

542 GENERAL NON-ZERO-SUM GAMES

A

motivation of that theory, as given in 56.2.2.-57.1. meets the latter require-
ment. If the reader reconsiders those sections in the light of the two pre-
ceding remarks, then he will see that the detailed justification in the desired
sense was their subject. Indeed, the possibility of the objection now
under consideration was our reason for making the discussion of our theory
so detailed, and avoiding plausible shortcuts. 1

Fourth: In spite of all this, the reader may feel that we have over-
emphasized the role of threats, compensations, etc., and that this may be a
one-sidedness of our approach which is likely to vitiate the results in
applications. The best answer to this is, as repeatedly pointed out before,
the examination of those applications.

We shall therefore consider definite applications which correspond to
familiar economic problems. Their study will disclose that our theory
leads to results which are, up to a certain point, in satisfactory agreement
with the usual common-sense views on these matters. This is the case as
long as the two following conditions are fulfilled: First that the setup is
simple enough to allow a purely verbal analysis, not making use of any
mathematical apparatus. Second that those factors which are inseparable
from our theory, but often excluded in the ordinary, verbal approach — coali-
tions and compensations — have not come essentially into play. This situ-
ation will be found to exist in the application of 61.2.2.-61.4. Indeed,
that example provides the decisive corroboration of our procedure.

Beyond this point, where the first condition is still satisfied, but not
the second, we shall find discrepancies just in the direction and to the
extent to which the difference in standpoint justifies it. This will be
particularly clear in the applications of 61.5.2., 61.6.3. and 62.6.

Finally, as even the first condition fails, because the problem is no longer
elementary, we gradually reach ground where the theoretical procedure
necessarily takes over the leading role from the ordinary, purely verbal
one. 2

59. General Considerations
59.1. Discussion of the Program

69.1.1. We can now proceed to the applications of our theory of the
general n- person game. The best way of starting such applications is a
systematic discussion of all general n-person games for small values of n.
It will appear that we can carry this out in absolute completeness for the
same n as for the zero-sum games: For the n ^ 3. The discussion for the
greater values, i.e. for n ^ 4, is necessarily at least as difficult as it was for

1 A possible one would have been to define the characteristic function as in 58.1. and
to come out then with a flat generalization of the theory for zero-sum games, i.e. with
(56:1 :d) in 56.12.

* This gradual transfer of the emphasis from corroboration of the theory by the
reliable common-sense results in the simple case, to overriding any untheoretical approach
by the theory in the complicated ones, is, of course, quite characteristic in the formation of
scientific theories.

GENERAL CONSIDERATIONS 643

the zero-sum games where we could only dispose of special cases of various
kinds.

We propose to do considerably less in the way of analyzing games with
n 4 this time. We can afford to be considerably briefer now than we
were in discussing the zero-sum games: The detailed discussion there was
necessary in order to reassure ourselves of the propriety of our procedure,
and of the general ideas and methodical principles underlying it. At the
stage which we have reached now, the general setup of the theory appears
to be justified, and we want only to gain assurance concerning the one
generalizing step carried out in this chapter. For this purpose a less exten-
sive analysis of applications should suffice.

Further, it will be possible already to connect the general games with
n £ 3 with some typical economic problems (bilateral monopoly, duopoly
versus monopoly, etc.) which allow judgment of the appropriateness of our
theory in the sense indicated before.

More detailed investigations of general games with n ^ 4 will be under-
taken in subsequent publications.

69.1.2. The systematic application of our new theory is best introduced
by a general discussion, similar to that of 31. It will not be necessary,
however, to carry out the equivalent considerations in detail; we must only
analyze to what extent the results obtained there carry over to the present
situation, or what modifications are required.

We need not discuss again the role of strategic equivalence, as expounded
in 31.3., since this subject has already been dealt with satisfactorily in
57.5.1. On the other hand, we shall take up certain matters originating
elsewhere than in 31.: reduced forms, inequalities which hold for the char-
acteristic function, inessentiality and essentiality (cf. 27.1.-27.5.); further,
the absolute values |r|i, |r| 2 (cf. 45.3.), and finally some remarks concerning
the theory of decomposition of Chapter IX.

69.2. The Reduced Forms. The Inequalities

69.2.1. The concept of strategic equivalence, as introduced in 57.5.1.
can be used to define reduced forms for all characteristic functions, along
the lines of 27.1.

Given a characteristic function v(S) its general strategically equivalent
transformation is given by (57:18) in 57.5.1., i.e. by

(59:1) v'OS) = v(S) + X «*•

kin S

This is precisely (27:2) in 27.1.1., but the a®, • • • , a* are now completely
unrestricted, while they were subject loc. cit. to the condition (27:1):

n

£ a* = 0. Hence the a?, • • • , a® are now n independent parameters,

Jk-l

544

GENERAL NON-ZERO-SUM GAMES

while they represented only n — 1 independent parameters formerly
(cf. 27. 1.3.). 1

It would be erroneous to assume, however, that this leads to more
restrictive possibilities of normalization than we found in 27.1.4. Indeed,
we desired loc. cit. to obtain a particular v'OS) — to be denoted by v(S ) —
which fulfills the n — 1 conditions (27 :3) :

(59:2) v((l)) = v((2)) » • • ■ = v((n)).

Yet, the characteristic functions considered at that time belonged to zero-
sum games; hence we had automatically

(59:3) v((l, •••,»))- 0.

In imposing this as a normalizing requirement, we now have n conditions:
(59:2) and (59:3). So we obtain

(59:4) v(J) + £ al = 0,

1

(59:5) v((l)) + o? = v((2)) + «§=•••= v((n)) + <*£•

(59:4) expresses (59:3); (59:5) expresses (59:2). These equations cor-
respond to (27:1*), (27:2*) loc. cit., and it is easy to verify that they are
solved by precisely one system of o?, • • • , a®:

n

(59:6) al = -v((*)) + l {2 v((*)) - v(7)J. S

*-l

So we can say:

(59 :A) We call a characteristic function v(S) reduced if and only if it

satisfies (59:2), (59:3). 3 Then every characteristic function
v(S ) is in strategic equivalence with precisely one reduced
vOS). This v(S ) is given by the formulae (59:1) and (59:6), and
we call it the reduced form of v($).

59 . 2 . 2 . Another possible requirement for the n parameters a®, • • • , aj|
consists in requiring for v'(£) — to be denoted by V(S) — the n conditions

(59:7) *((1)) = v((2)) = • • • = ♦((»)) - 0.

1 Our present standpoint in this respect is similar to that which we took for the
constant-sum games in 42.2.2.

* Proof: Denote the joint value of n terms in (59:5) by /3. Then (59:5) amounts to

n

«* “ — v((fc)) + 0 and so (59:4) becomes v(/) — ^ v((fc)) + n/3 - 0, i.e.

fc-i

j ji v«fc)> -

1 This is precisely the definition of 27.1.4.

GENERAL CONSIDERATIONS

545

This means

(59:8) v((l)) + a? = v((2)) + «*=•••« v((n)) + = 0,

i.eu

(59:9) al = -v((fc)).

So we can say:

(59 :B) We call a characteristic function v(S) zero reduced if and only

if it satisfies (59:7). Then every characteristic function v(S)
is in strategic equivalence with precisely one zero-reduced v($).
This v(S) is given by the formulae (59:1) and (59:9), and we
call it the zero-reduced form of v(S).

59.2.3. Let us consider the reduced characteristic function v(S). We
denote the joint value of the n terms in (59:2) by — 7 , i.e.

(59:10) -7 - v((l)) - v((2)) = • • • = v((n)).

Hence — 7 = v((fc)) + «£ and so (59:6) gives

n

(59:11) y = l jv(Z) - £v((fc))J-

Jk-1

If we use the zero-reduced form v(£) of the same v(S) then we have

n n

v(/) = v(J) + X hence by (59:9) f(I) = v(/) - £ v((fc)), i.e. using

jfc-1 ib-1

(59:11)

(59:12) ny = ^(/).

Returning to the reduced form v(S), we see that some equalities and all
inequalities of 27.2. are still valid.

To begin with, (59:10) can be stated as follows:

(59:13) v{S) — — 7 for every one-element set S.

This coincides with (27:5*) loc. cit., while (27:5**) id. fails, since we saw
in 57.2.1. that the equivalent of (25:3:b) in 25.3.1. is now missing, and this
was required to derive (27:5**) from (27:5*) there.

Repeated application of (57:2:c) in 57.2.1. to the sets ( 1 ), • • • , (n)
gives by (59:13) — ny ^ 0, i.e.:

(59:14) 7 S 0.

This coincides with (27 :6) in 27.2.

Consider next an arbitrary subset S of I. Let p be the number of its
elements: S = (Au, • • • , k p ). Repeated application of (57:2:c) in 57.2.1.

646

GENERAL NON-ZERO-SUM GAMES

to the sets • • • , (k p ) gives by (59:13)

vOS) ^ -P7.

Apply this to — $ which has n — p elements. Owing to (57:2:b) in 57.2.1.
and (59:3), we have

v( — S) ^ -vOS) 1

hence the preceding inequality now becomes

v(S) g (n - p)y.

Combining these two inequalities gives :

(59:15) — py ^ v(/S) ^ (n — p)y for every p-element set £.

This coincides with (27:7) in 27.2.

(59:13) and v(Q) = 0 (i.e. (57:2:a) in 57.2.1.) can also be formulated
as follows:

(59:16) For p — 0, 1 we have = in the first relation of (59:15).

This coincides with (27:7*) in 27.2. v(7) = 0 (i.e. (59:3)) can also be

formulated as follows :

(59:17) For p = n we have = in the second relation of (59:15).

This coincides with (27:7**) loc. cit., except that p = n — 1 is missing, for
the same reason for which the equivalent of (27:5**) id. is missing (cf. the
remark following our (59:13)).

59.3. Various Topics

59 . 3 . 1 . These inequalities can now be treated in the same way as in
27.3.1.

There are two alternatives, based on (59:14):

First case: y = 0. Then (59:15) gives v(S) = 0 for all S. This is
precisely the inessential case discussed in 27.3.1., with all the attributes
enumerated there. Considering (59 :A), the inessential games are precisely
those which are equivalent to the game with v(S) = 0, — the game which is
perfectly “ vacuous/'

Second case: 7 > 0. By a change of unit we could make 7 = 1, with
the consequences pointed out in 27.3.2. And just as there, we refrain
from doing this immediately. For the same reasons as pointed out there,
the strategy of coalitions is decisive in such a game. We call a game in this
case essential

The criteria (27 :B), (27 :C), (27 :D) of 27.4. for inessentiality and essenti-

n

ality are again valid: In (27 :B) £ v((fc)) must be replaced by

*~i

1 Note that in our present application this inequality replaces this missing equality
(25:3:b) in 25.3.1., which was used in 27.2.

GENERAL CONSIDERATIONS

547

2 v ((*0) - v ( 7 )»
fc -1

while (27 :C), (27 :D) are completely unaffected. Indeed, it is easy to
verify that the proofs given there carry over to the present case, their bases
being provided in 59.2.1.

We leave it to the reader to apply the considerations of 27.5. — for the
essential case, with the normalization y = 1 — to the present situation.

59.3.2. We can now pass to the considerations which correspond to
those of 31.

The remarks of 31.1.1.-31.1.3. concerning the structure of the concept of
domination and certainly necessary and certainly unnecessary sets can be
repeated without any change. The concepts of convexity and of flatness
can be introduced as in 31.1.4. The conclusions of 31.1.4.-31.1.5. are also
unaffected, except for (31:E:b) in 31.1.4. and (31 :G) in 31.1.5., as well as
(31 :H) id. for p = n — 1. These are the only ones where (25:3 :b) of 25.3.1.
(cf. 57.2.1.) is used.

Finally, the remark at the end of 31.1.5. must be modified. Owing to
what we said above, the value p = n — 1 is just as dubious as those included
in (31:8) loc. cit. I.e. the p for which the necessity of S is in doubt, are
restricted to p 0, 1, n, i.e. to the interval

(59:18) 2 g p £ n - 1.

Thus this interval begins to play a role when n ^ 3, — not only, as loc. cit.,
when n ^ 4. 1

Consider next the results of 31.2. The reader who consults that sec-
tion will have no difficulty in verifying the following: (31:1), (31 :J), (31 :K)

— > — >

are unaffected. In (31 :L) the construction of p with the help of a can

be carried out without any change; the first assertion, P H a , cannot be
maintained, since it uses that part of (31 :H) in 31.1.5. which is no longer

valid; the second assertion, not a h p , is unaffected. This weakening
of (31 :L) removes (31 :M). (31 :N) remains true because it uses the intact

part of (31 :L) only. (31:0), (31 :P) are unaffected.

59.3.3. To conclude, let us consider some of the concepts of Chapter IX.
We defined there the two numbers |r|i, |r|* the former in 45.1., the latter

in 45.2.3., and we discussed their properties in 45.3.

Both definitions — i.e. the pertinent considerations of 45.1., 45.2. —
carry over literally. There are, however, essential changes in 45.3.: In
(45 :F) only the second part of the proof is valid, but not the first part,
since that — and that alone — makes use of (25:3:b) in 25.3.1. (cf. 57.2.1.).

1 This is in agreement with the connection of general n-person games and sero-sum
n + 1-person games, which was prominent throughout 56.2-56.12.

548

GENERAL NON-ZERO-SUM GAMES

Specifically: We still have

(59:19) |r| s g-~|r|„

and so we can evaluate |r|* in terms of |r|i; but we do not have
(59:20) |r|i ^ (n - l)|r| 2 , 1

nor can we evaluate |r|i in terms of |r| 2 at all. Indeed, we shall see in 60.2.1.
that

(59:21) \TU > 0, |r|, = 0

occurs for certain games.

In consequence of this, the remarks of 45.3.3.-45.3.4. become pointless.
The same holds for 45.3.1., i.e. its result (45 :E) fails in so far as it concerns
|r| t . It is true for |r|i but this is merely a restatement of the definitions.
Considering this, and (59:19), (59:21) above, we see that (45 :E) must be
weakened as follows:

(59 :C) If T is inessential, then |r|i = 0, |r|* = 0.

If T is essential, then |T|i > 0, |r| 2 ^ 0.

The theory of composition and decomposition , which is the main object
of Chapter IX, can be extended in its essential parts to our present set-up.
The difference between the behavior of |T|i and |r| 2 discussed above, neces-
sitates some minor changes, but these are easily applied. Of course the
theory of excesses and of solutions in the sets E(e 0 ) and F(e 0 ) (cf. there) must
be extended to the present case — but this, too, entails no real difficulties.

A detailed analysis of this subject would lengthen our exposition beyond
the limits that we set ourselves in 59.1.1. Furthermore, the interpretative
value of the results would not differ materially from what was already
obtained in Chapter IX, when considering zero-sum games.

60. The Solutions of All General Games with n ^ 3

60.1. The Case n - 1

60.1. We proceed to the systematic discussion of all general n-person
games with n ^ 3, as announced in 59.1.1.

Consider first n = 1. This case has already been considered (and, for
practical purposes, settled) in 12.2. In particular, we pointed out in 12.2.1.
that in this (and only in this) case we deal with a pure maximum problem.
It is nevertheless desirable to verify that our general theory produces in this
(trivial) special case the common-sense result. 2 We apply therefore the
general theory in complete mathematical rigor.

1 (59:20) and (59:19) express the two parts of (45 :F), respectively.

* This brings us back to the fourth remark in 58.3.2.

SOLUTIONS FOR n g 3

549

A general game T with n = 1 is necessarily inessential: This is clear by
considering the characteristic function v(S) of its reduced form, since
then (59:16) and (59:17) in 59.2.3. give (for p = 1 = n) — 7 = 0, i.e. 7 = 0.
We may also use — without reducing — any one criterion (27 :B), (27 :C),
(27:D) of 27.4. (cf. 59.3.1.). E.g. (27 :C) loc. cit. is clearly satisfied, with
an = v((l)). Note that this is v(7), i.e. by (56:13) in 56,9.1. (with the
notation of 12.2.1.) Max T 3C(r). We restate this:

(60:1) ai = v((l)) = v(Z) = Maxr 3C(r).

Since T is inessential, we can apply (31:0) or (31 :P) in 31.2.3. (cf. 59.3.2.).
This gives:

(60: A) T possesses precisely one solution, the one-element set (a)

where

a = { {ati } }
with the ai of (60:1).

This is obviously the “ common-sense” result of 12.2.1. — as it should be.

60.2. The Case n — 2

60 . 2 . 1 . Consider next n = 2. The main fact is that a general game with
n = 2 need not be inessential — thus differing from the zero-sum games with
n = 2. (The latter are inessential by the first remark in 27.5.2.)

Indeed : The characteristic function v(S) of its reduced form is completely
determined by (59:16) and (59:17) in 59.2.3. It is

(60:2)

v(S)

0

— 7 when S has
0

0

1 elements.

2

Now one verifies immediately that a v(S) of (60:2) fulfills the conditions
(57:2:a), (57:2:c) of 57.2.1., i.e. that it is the characteristic function of a
suitable T (cf. 57.3.4.), if and only if 7 ^ 0. This is precisely the condition
(59:14) in 59.2.3. So we see: the 7 ^ 0 of (59:14) in 59.2.3. are precisely
the possibilities in (60:2).

Thus 7 > 0, i.e. essentiality, is among the possibilities, as asserted.
In the case of essentiality we may further normalize 7 = 1, thereby com-
pletely determining (60:2). Thus there exists only one type of essential
general two-person games.

Note that while |T|i = 27 may thus be > 0, there is always (for n = 2)
|r|, = 0. It suffices to prove this for the reduced form, i.e. for (60:2).

Indeed: Recalling the definitions of 45.2.1. and 45.2.3. we see that

a = { {ct h a 2 } ) is detached when an, a* ^ — 7, an + a* ^ 0, and that the
minimum of the corresponding e = an + a* is 0. 1 Hence |r|* = 0 as
desired.

1 It is assumed e.g. for ai * at — 0.

560

GENERAL NON-ZERO-SUM GAMES

Summing up: For n = 2 a zero-sum game must be inessential, a general
game need not be. Accordingly the former must have |r|i = 0; the latter
may have |T|i > 0 too. But both have always | F| 2 = 0.

We leave it to the reader to interpret this result in the light of previous
discussions, and particularly of 45.3.4.

60 . 2 . 2 . The solutions for a general game T with n = 2 are easily deter-
mined.

By the valid part of (31 :H) in 31.1.5. (cf. the pertinent observations
in 59.3.2.) all sets S £ I with 0, 1 or n elements are certainly unnecessary —
but since n = 2, these exhaust all subsets. Hence we may determine the
solutions of T as if domination never held. Consequently a solution is
simply defined by the property that no imputation can be outside of it.
I.e. there exists precisely one solution: the set of all imputations.

The general imputation is given in this case as a = { {au, a 2 } }, subject
to the conditions (57:15), (57:16) in 57.5.1., which now become:

(60:3) «i ^ v((l)), a 2 ^ v((2)),

(60:4) ai + a 2 = v((l,2)) = v(J).

We restate the result :

(60 :B) T possesses precisely one solution, the set of all imputations.

These are the

— >

a = { {oti, a 2 | |

with the ai, a 2 of (60:3), (60:4).

Note that (60:3), (60:4) determine a unique pair an, a 2 (i.e. a ) if and
only if

(60:5) v((l)) + v((2)) = v((l,2)).

By the criteria of 27.4. this expresses precisely the inessentiality of T.
This result is, as it should be, in harmony with (31 :P) in 31.2.3. (cf. 59.3.2.).
Otherwise

(60:6) v((l)) + v((2)) < v((l,2)),

and there exist infinitely many a i, a 2 , — i.e. a . This is the case of essential-
ity for r.

The interpretation of these results will be given in 61.2.-61.4.

60.3. The Case n » 3

60 . 3 . 1 . Consider finally n = 3. These games include the essential
zero-sum three-person game for which |r|i > 0 and |r|* > 0 (cf. 45.3.3.).
So we see :

For n = 3 a zero-sum game as well as a general game may be essential,
and both |r|i >0 and |r| 2 > 0 ate possibilities.

SOLUTIONS FOR n £ 3

551

The case where T is inessential is taken care of by (31:0) or (31 :P) in
31.2.3. (cf. 59.3.2.). We assume therefore that T is essential.

Use the reduced form of T in the normalization 7 = 1. Then we
can describe its characteristic function v(S) with the help of (59:16) and
(59:17) in 59.2.3. as follows:

f 0

(60:7) v(iS) = | -1

l 0

and

(60:8) v((2,3)) = a h v((l,3)) = a 2 , v((l,2)) = a 3 when S has

2 elements.

And it is verified immediately that a v(S) of (60:7), (60:8) fulfills the con-
ditions (57:2:a), (57:2:c) of 57.2.1., i.e. that it is the characteristic function
of a suitable T (cf. 57.3.4.), if and only if

(60:9) -2 go,, a 2 , a 3 ^ 1.

Note that this Y can be chosen zero-sum, i.e. that (25:3:b) of 25.3.1.
holds if and only if

(60:10) ai = a 2 = «3 = 1.

In other words: The domain (60:9) represents all general games, while its
upper boundary point (60:10) represents the (unique) zero-sum game of
our case.

60 . 3 . 2 . Let us now determine tlfle solutions of this (essential) general
three-person game.

' — y

The general imputation is given in this case as a = { {an, c* 2 , «s} },

subject to the conditions (57:15), (57:16) in 57.5.1., which now become:

(60:11) «i = — 1, «2 = — 1, a 3 ^ - 1,

(60:12) a\ a 2 + a 3 = 0.

These conditions are precisely those of 32.1.1. for a h a 2 , <x 3 (cf. (32:2),
(32:3) there), i.e. those used in the theory of the essential zero-sum three-

person game. They agree also, apart from the factor 1 + with the

conditions of 47.2.2. for a 1 , a 2 , a 3 (cf. (47:2*), (47:3*) there), i.e. with those
used in the theory of the essential zero-sum three-person game with excess.

Consequently we can use the graphical representation described in 32.1.2.,

— >

in particular in Figure 52. We obtain the domain of the a as the funda-
mental triangle in 32.1.2. in Figure 53. It is also similar to that in 47.2.2.
in Figure 70.

We express the relationship of domination in this graphical representa-

— > ►

tion. Concerning the set <S of 30.1.1. for a domination a s- /3 , the follow-

552

GENERAL NON-ZERO-SUM GAMES

ing can be said. By the valid part of (31 :H) in 31.1.5. (cf. the pertinent
observations in 59.3.2.) all sets S £ I with 0, 1 or n elements are certainly
unnecessary — but since n = 3, this restricts our analysis to two-element
sets S.

Put therefore S = (t, j). 1 Then domination means that

on + otj ^ v((t, j)) = a* and a, > ft, a, > ft.

By (60:12) the first condition may be written a k ^ — a k .

We restate this: Domination

a H 0

means that

f either a 1 > /Si,

CK2 > j9l

and

«3 =

-a z ;

(60:13)

or an > 01,

a 3 > 03

and

a 2 =

— a 2 ]

1 or a 2 > 0 S ,

a 3 > 03

and

ai ^

~Ul . 2

The circumstances described in (60:13) can now be added to the picture
of the fundamental triangle. The similarity is now more with 47. than with
32. The operation corresponds to the transition from Figure 70 to Figures
71, 72, or to Figures 84, 85, or to Figures 87, 88. Indeed, the difference as
against Figures 71, 84, 87 (which all describe the same operation, in the
successive Cases (IV), (V), (VI)) is only this:

The six lines

which form the configuration there, are now replaced by the six lines

(60:15)

Oil =

-i,

a 2 —

-i,

«3

on =

—a i,

Oil =

02,

otz

respectively. Hence the second triangle (formed by the three last lines)
which appears in the fundamental triangle (formed by the three first lines)
need not be placed symmetrically with respect to the latter, as it is in the
three figures mentioned.

1 i, j , k a permutation of 1,2,3.

2eo

* This is quite similar to (47:5) in 47.2.3., except that we have there 1 in place

£o

of all three a if a*, a*. There is also the change of scale by the factor 1 -f referred to
after (60:11), (60:12).

The relation to (32:4) in 32.1.3. is the same as for (47:5) in 47.2.3., cf. footnote 2
on p. 406.

SOLUTIONS FOR 3 553

60 . 3 . 3 . It is convenient to distinguish two cases, according to whether
the

(60:16) a i ^ — ai, <*2 = — <* 2 , <*z S —ns

sides of the three last lines of (60:15) (where the three domination relations
of (60:13) are valid) intersect in a common area, or not. Owing to (60:12)
the former means that

(60:17:a) fli + o>z > 0,

while the latter means that

(60:17:b) <Zi -f- CL 2 + dz ^ 0.

We call these cases (a) and (b), respectively.

Case (a): We have the conditions of Figures 71, 72, except that the
inner triangle need not be placed symmetrically with respect to the funda-
mental triangle, as it is there. If this is borne in mind, then the discussion
of Case (IV), as given in 47.4.-47.5. can be repeated literally. The solutions
are therefore, with the same qualification, those depicted in Figures 82, 83.

We note that if an a< = 1, then the corresponding sides of the inner
and the fundamental triangle coincide (cf. (60:15)), and the corresponding
curve disappears. 1

Case (b): We have essentially the conditions of Figures 84, 85 — of
which those of Figures 87, 88 are but a variant — with the same proviso for
asymmetry as in Case (a) above.

We redraw the arrangement of Figure 84, the fundamental triangle
being marked by / and the inner triangle by \: Figure 92. The arrange-
ment has several variants, because the inner triangle can stick out from
the fundamental triangle in various ways. 2 Figures 92-95 depict these
variants. 8 * 4

If these circumstances are borne in mind, then the discussion of Case (V)
as given in 47.6. can be repeated literally. 8 The solutions are therefore,

1 Thus in the zero-sum case, where ai =03 =* a% ■* 1, none of these curves occur —
in accord with the result of 32.

2 By (60:9) — 2 £ a,- ^ 1. This means, as the reader may easily verify for himself,
that each side of the “inner” triangle must pass between the corresponding side of the
fundamental triangle and its opposite vertex. Our Figures 92-95 exhaust all possibilities
within this restriction.

* The only ones which can occur in a zero-sum game, i.e. for ai — at » as — 1, are
those which can be symmetric: Figures 92, 95. Of these, Figure 92 corresponds to
Figure 84, and Figure 95 corresponds to Figure 87.

4 The Figures 92-95 differ from each other by the successive disappearance of the
areas ®, ®, ®. Besides one or more of the areas ® and ®, ®, ® may degenerate to a
linear interval or even to a point. It is sometimes not quite easy to distinguish between
the “disappearance” mentioned above, and this “degeneration.” A rule which allows
differentiation between the four cases corresponding to the Figures 92-95 and in which
this difficulty does not present itself, is this: Figures 92-95 correspond respectively to
the cases where the “inner” triangle meets 0, 1, 2, 3 sides of the fundamental triangle.
(Meeting a vertex counts as meeting both sides to which it belongs.)

6 The discussion of Case (VI) in 47.7. may also be considered as such a repetition —
under much simpler conditions.

554

GENERAL NON-ZERO-SUM GAMES

with the necessary qualifications of asymmetry and the possible disappear-
ance or degeneration of some areas 0-® (cf. Figures 92-95 and footnote
4 on p. 553), those depicted in Figure 86.

Figure 95.

60.4. Comparison with the Zero-sum Games

60 . 4 . 1 . We have determined all solutions of the general n-person games
with n = 3 in a rigorous way, but we have not yet made an attempt to
analyze the meaning of our results. We therefore pass now to this
analysis.

Let us begin with some remarks of a rather formal nature. We have
seen that the smallest n for which a general game can be essential is n = 2,
while for the zero-sum games the corresponding number was n = 3. We
have also seen that there exists (assuming reduction and normalization
7 = 1) precisely one essential general game for n = 2, whereas for the
zero-sum games the same thing was true for n = 3. Again the essential
general games for n = 3 (under the same assumptions as above) form a
three parameter manifold, while for the zero-sum games this was true for
n = 4. All this indicates an analogy between general n- person games and
zero-sum n + 1-person games. Of course, we know the reason: The zero-
sum extensions of the general n-person games are zero-sum n + l-person

ECONOMIC INTERPRETATION FOR n = 1,2 555

games and we saw that every zero-sum n + 1 -person game can be obtained
in this way. 1

60.4.2. It must be remembered, however, that while the zero-sum
n + 1-person games are exhausted by this procedure, their solutions are
not — the solutions of a general n-person game form only a subset of those
of its zero-sum extension (cf. e.g. (56 :1 :a) in 56.12.).

Thus our determination of all solutions of ‘all general three-person games
means only that we know some, but not all solutions of all zero-sum four-
person games. Indeed, the voluminous and yet incomplete discussion of
Chapter VII shows that determining all solutions of all zero-sum four-person
games is a task of considerably greater size. Our results concerning the
general three-person games imply, however, this much: There exist solu-
tions for every zero-sum four-person game. (The casuistic discussion of
Chapter VII did not reveal this.)

61. Economic Interpretation of the Results for n = 1,2
61.1. The Case n = 1

61.1. We now come to the main objective of our present analysis: The
interpretation of our results for n = 1,2,3.

Consider first n = 1 : What matters in this case was already stated or
referred to in 60.1. Our result was, as it had to be, a repetition of the simple
maximum principle which characterizes this case — and this case only,
which therefore describes the “ Robinson Crusoe” or completely planned
communistic economy.

61.2. The Case n - 2. The Two-person Market

61.2.1. Consider next n = 2: Our result for this case, obtained in 60.2.2.
can be stated verbally as follows :

There exists precisely one solution. It consists of all those imputations
where each player gets individually at least that amount which he can
secure for himself, while the two get together precisely the maximum amount
which they can secure together.

Here the “ amount which a player can get for himself” must be under-
stood to be the amount which he can get for himself, irrespective of what
his opponent does, even assuming that his opponent is guided by the desire
to inflict a loss rather than to achieve a gain. 2

In examining the solution we find the opportunity to fulfill the promise
contained in the fourth remark in 58.3.2.: We must see whether our above
definition of the “ amount which a player can get for himself based on a
hypothetical desire of the opponent to inflict a loss rather than to achieve

1 Precisely : It is strategically equivalent to one which is so obtainable. (Cf . the
beginning of 57.4.1.)

* Cf. the detailed discussion at the end of 58.2.1. and in 58.3. The amount which the
player k can secure for himself is, of course, v((fc)).

556

GENERAL NON-ZERO-SUM GAMES

a gain — leads to common-sense results. 1 In order to compare the result
of our theory in this way with “common-sense,” it is desirable to present
the general two-person game in a form which is easily accessible to ordinary
intuition. Such a form is readily found by considering some fundamental
economic relationships which can exist between two persons.

61 . 2 . 2 . Accordingly we consider the situation of two persons in a market,
a seller and a buyer. We wish to analyze one transaction only and it will
appear that this is equivalent to the general two-person game. It is
obviously also equivalent to the simplest form of the classical economic
problem of bilateral monopoly.

The two participants are 1,2: the seller 1 and the buyer 2. The
transaction which we consider is the sale of one unit A of a certain commod-
ity by 1 to 2. Denote the value of the possession of A to 1 by u and for 2
by v. I.e., u represents the best alternative use of A for the seller, while v
is the value to the buyer, after the sale.

In order that such a transaction have any sense, the value of A for the
buyer must exceed that one for the seller. I.e. we must have

(61:1) u<v.

It is convenient to use the state of the buyer when no sale occurs — i.e.
his original financial position — as the zero of his utility. 2

Let us now describe this as a game. In doing this it is best to omit A
from the picture altogether and to deal instead with the value connected
with its transfer or its alternative uses. We may then formulate the rules
of the game as follows.

1 offers 2 a “price” p, which 2 may “accept” or “decline.” In the
first case 1, 2, get the amounts p, v — p. In the second case they get the
amounts u> 0. 3

The common-sense result is that the price p will have some value
between the limits set by the alternative valuations of the two participants,

1 The reader will understand that we do not ascribe this desire to the opponent. It
is only that our theory can be formulated as if he had this desire. What matters is not
this possible formulation, but the results of the theory.

Indeed, this “malevolent” behavior of the opponent determines only some, but
not all features of the solution : It gives the lower limit of what each player must obtain
individually, but what both get together can only be described by the opposite hypothesis
of perfect cooperation. (Cf. above.)

This is just a special case of the general fact, that only the entire, rigorous theory is
a reliable guide under all conditions, while the verbal illustrations of its parts are of
limited applicability and may conflict with each other.

All this can be brought out even better by the detailed discussion of 58.3.

* We are purposely disregarding the possibility of describing a sale as an exchange of
goods for goods. Our theory forces us, for reasons which we have stated repeatedly, to
use an unrestrictedly transferable numerical utility, which we may as well describe in
terms of money.

We shall deviate from this standpoint only in Chapter XII.

1 We leave it to the reader to formulate this in terms of our original combinatorial
definition of games.

ECONOMIC INTERPRETATION FOR n = 1,2

557

e. that

51:2) u ^ p ^ v .

inhere p will actually be between the limits of (61:2) depends on factors
ot taken into account in this description. Indeed, this rule of the game
rovides for one bid only, which must be accepted or declined — this is
learly the final bid of the transaction. It may have been preceded by
egotiating, bargaining, higgling, contracting and recontracting, about
r hich we said nothing. Consequently a satisfactory theory of this highly
implified model shoulcHeave the entire interval (61:2) available for p.

61.3. Discussion of the Two-person Market and Its Characteristic Function

61 . 3 . 1 . Before going any further we add two remarks concerning this
escription of the game, which is our model for the economic set-up under
onsideration.

First: It would be possible to use more elaborate models allowing for
reater (but limited) numbers of alternative bids, etc.

There is a prima facie evidence for considering such variants, since all
listing markets are governed by more or less elaborate rules for successive
ids by all participants, which appear to be essential for the understanding
f their character. Besides, we did investigate in detail the game of
oker in 19. This game is based on the interplay of the bids of all partici-
ants and we saw loc. cit. that the sequence and arrangement of these bids
as of decisive importance for its structure and theory. (Cf. in particular
re descriptive part 19.1.-19.3., the variants discussed in 19.11.-19.14. and
le concluding summary of 19.16.)

A closer inspection shows, however, that in our present setup these
etails do not become decisive. The situation is altogether different from
oker which is a zero-sum game and where any loss of one player is a gain
>r the other one. 1 Specifically, the reader may discuss any more compli-
stted market (but with only two participants !) in the same way as we shall

0 it for our simple version in 61.3.3. He will find the same characteristic
mction as we obtain (61 :5), (61 :6) of 61.3.3. Indeed, the deductions given
tiere apply mutatis mutandis in any market (of two participants!): The
jader who carries out this comparison will observe that all that matters

1 those proofs 2 is that the seller (or the buyer) may, if he wishes, insist
bsolutely on the particular price mentioned there, irrespective of the
ounter-offers he may get and the number of successive bids required. 2

These elaborations lead essentially to the same results as our simple
lodel. We refrain therefore from considering them.

1 This applies directly to Poker as a two-person game, as considered in 19. If more
lan two persons participate, then our treatment by means of coalitions brings about the
ime situation.

2 The significant one is the proof of (61:5) in 61.3.3.

* Returning to our previous remarks concerning Poker: The reader may verify for
imself how a corresponding simple overall policy would not work there — due to the

558

GENERAL NON-ZERO-SUM GAMES

61 . 3 . 2 . Second: On the other hand our mode! could also be simplified
further. Indeed, the mechanism of compensations between (co-operating)
players, which we assumed in all parts of our theories is perfectly adequate
to replace bids of prices. I.e. it is not necessary to introduce offering,
accepting or declining of prices as part of the rules of the game. The
mechanism of compensations is fully able to take care of this, including the
preliminary negotiating, bargaining, higgling, contracting and recontracting.

Such a simplified game could be described as follows: Both players 1,2
may choose to exchange or not. If either one chooses not to exchange, then
1,2 get the amounts u, 0. If both chose to exchange then they get the
amounts w', u " — where u', u " are two arbitrary but fixed quantities with
the sum v. 1

In other words: The rules of the game may provide for an arbitrary
“price” p = u' (then v — p = u"), which the players cannot influence —
they will nevertheless bring about any other price they desire by appropriate
compensations.

Thus it appears that the arrangement chosen in 61.2.2. is neither the
simplest nor the most complete one. We are using it because it seems to be
best suited to bring out the essential traits of the situation without unneces-
sary details.

61 . 3 . 3 . The “common-sense” result of 61.2.2. amounts in the terminol-
ogy of imputations to this: There exists precisely one solution and this is the
set of all imputations

a = ( {on, a 2 }},

with

(61:3) «i ^ u, «2 ^ 0,

(61:4) «i + a 2 = v .

Comparing this with the application of our theory in 60.2.2., we see
that agreement obtains when (61:3), (61:4) coincide with (60:3), (60:4)
there. This means that we must have

(61:5) v((l)) = u, v ((2)) = 0,

(61:6) v((l,2)) = i>.

It is easily verified that (61 :5), (61 :6) are indeed true. For the sake of
completeness we do this for both arrangements of 61.2.2. and 61.3.1.,

penalties which the rules of that game inflict upon any prohibitive, excessive, or in any
other simple way uniform scheme of bidding.

One could, of course, incorporate similar provisions into the rules governing a
market. Indeed, there are certain traditional forms of transactions which are possibly
of this type, such as options. But it does not seem advisable to include them in this
first, elementary survey of the problem.

1 The characteristic functions of both arrangements (that of 61.2.2. and the one above)
will be determined in 61.3.3. and they will be found to be identical.

ECONOMIC INTERPRETATION FOR n = 1,2 559

61.3.2., the first being dealt with in the text, and the variants required for
the second alternative in brackets [ ].

Ad (61:5): Player 1 can make sure to obtain u by offering the price
V = u [by choosing not to exchange]. Player 2 can make sure that player 1
obtains u by declining every price [by choosing not to exchange]. Hence
v((l)) = u .

Replacement of p = u by p = v [the same conduct of both players]
yields in the same way that v((2)) = 0.

Ad (61 :6): The two players together get either u or v — the latter arising
from p + (v — p) [from u' + w"]. By (61:1) v is preferable; hence

v((l,2)) = t;.

61.4. Justification of the Standpoint of 68.

61 . 4 . The coincidence of the values of the characteristic function v(S)
with the u , 0, v as observed in 61.3.3. may appear fairly trivial. There is,
however, one significant point about it: It was obtained with our definition
of the characteristic function to which the criticisms of 58.3. and 61.2.
apply. I.e. it is dependent upon each player ascribing to his opponent — in
a certain part of the theory but not in all of it — the desire to inflict a loss
rather than to achieve a gain.

It is important to realize that this dependence is really significant,
i.e. that modification of this assumption would alter the result, and therefore
falsify it, since the result was seen to be correct. This is best done with the
arrangement of 61.2.2.

Indeed, assume that player 2 would under certain conditions prefer
to make a profit for himself rather than to inflict a loss upon player 1.
Assume that these conditions exist, e.g. when player 1 offers a certain price
p 0 > u but < v. In this case player 2 obtains v — p 0 if he accepts, and 0
if he declines. Hence he gains by accepting. On the other hand player
1 obtains p 0 if player 2 accepts and u if he declines. Hence player 2 inflicts
a loss (upon player 1) by declining. Consequently our present assumption
concerning the intentions of player 2 means that he will accept.

Thus under these conditions player 1 can count upon obtaining the
amount p 0 . This conflicts with our previous result according to which the
entire price interval (61:2) should be permissible, and we saw in 61.2.2. that
it is the latter result that must be considered the natural one.

Summing up: The discussion of the general two-person game which we
carried out in 61.2.-61.4. has shown that the general two-person game is
crucial with regard to the decision whether the characteristic function
should be formed as used in our theory. The setup was simple enough as
to allow a common-sense” prediction of the result — and any change in
the procedure of forming the characteristic function would have altered
the theoretical result significantly. In this way we have obtained by the
application of the theory, a corroboration in the sense of the fourth remark
in 58.3.

560

GENERAL NON-ZERO-SUM GAMES

61.6. Divisible Goods. The “ Marginal Pairs”

61 . 5 . 1 . The discussion of 61.2.-61.4. referred to a very elementary case
but it nevertheless sufficed for the task of “corroboration” which we had
set for ourselves. Besides, by interpreting one essential general two-person
game, all were interpreted, since all of them are strategically equivalent
to one reduced form (which could be normalized to y = 1).

So far everything is satisfactory. But it is still desirable to verify that
our theory can do equal justice to somewhat less trivial economic setups.
For this purpose we will first extend the description of the two-person market
somewhat. It will be seen that this yields nothing really new. Then we
shall turn to the general three-person games. There we will find genuinely
new corroborations and opportunities for more fundamental interpretations.

61 . 5 . 2 . Let us return to the situation described in 61.2.2: the seller 1
and the buyer 2 in a market. We allow now for transactions involving
any or all of s (indivisible and mutually substitutable) units Ai, • • • , A,
of a commodity. 1 Denote the value of the possession of t(— 0, 1, • • • , s)
of these units for 1 by u t and for 2 by v t . Thus the quantities

(61 :7) uq = 0, Mi, • • • , m„

(61:8) v 0 = 0, t>i, * * * , v„

describe the variable utilities of these units to each participant. As in

61.2.2. we use for the buyer his original position as the zero of his utility.

There is no need to repeat the considerations of 61.2.2., 61.3.1., 61.3.2.
concerning the rules of the game which models this setup.

It is easy to see, what its characteristic function must be. Since each
player can block all sales, 2 it follows as in 61.3.3. that

(61:9) v((l)) = u t} v((2)) = 0.

Since the two players together can determine the number of units to be
transferred and since with a transfer of t units they obtain together u,- t + v t)
therefore

(61:10) v((l,2)) = Max M ,i . (u a - t + v t ).

This v(S) is a characteristic function, hence it must fulfill the inequali-
ties (57:2:a), (57:2:c) of 57.2.1. Considering (61:9), (61:10), the only one
which is not immediately obvious is

(61:11) v((l,2)) £ v((l)).

This obtains, by observing that the left-hand side is §: u a + vq = u a by
(61:10) (use t = 0), and the right-hand side is = u a .

1 We could also allow for continuous divisibility, but this would make no material
difference.

* 1 by offering an inacceptibly high price, 2 by declining every price.

561

ECONOMIC INTERPRETATION FOR n « 1,2

61 . 5 . 3 . Consider now the t for which the maximum in (61 :10) is assumed,
say t = i 0 . It is characterized by u 9 -i o + u a -t + v t for all t . This
need only be stated for the t to and we can state it for the t ^ to separately.
We may write these inequalities as follows:

(61:12) u a -t 9 — u a -t ^ v t — v h for t > t 0 ,

(61 :13) u a -t — u,- t ^ v t — v t for t < t 0 .

0 0

Specialize (61:12) to t = t 0 + 1 (except when to = s in which case (61:12)
is vacuous) :

( 61 : 14 ) u 9 - to — u ,- t - 1 ^ vt 9 + 1 - v to ,

and (61:13) to t = to — 1 (except when t 0 = 0 in which case (61:13) is
vacuous) :

(61:15) + 1 - ^ v t —

0 0 0 0

Note that (6:12), (6:13) (without the specialization t = / 0 i 1 that led to
(6:14), (6:15)) can be written as follows

t t

(61 :16) £ (w.-,+i - ?<,_,) S V ( Vj - Vj _,) for l > U>,

» — <o+l J - <0 + 1

*0 *0

(61:17) (*'*-+1 “ w.-.) ^ fa - v,-i) for f < i a .

In general we can say that (61:14), (61:15) is necessary only, while
(61:16), (61:17) is necessary and sufficient. However, we may now profit-
ably introduce the assumption of decreasing utility — that is, that the utility
of each additional unit decreases, as the total holding increases, for both
participants 1,2. As a formula

(61:18)

(61:19)

U\ — Uq > ll* — U\ > * • • > U 8 — Us- 1 ,

r i — r 0 > V2 — v\ > • * ■ > v t — v 9 - 1.

This implies

(61:20)

2} (w.-.+l - W«— i) ^ (/ - t o)(m.-(, - l)j

I *“<0 + 1 *

t

v (V, - ~ lo)(t\+i - t\)

*0

5) (w—.+i - w—,) ^ (to — o(«.-i 0 +i -

to

X ( v >~ ”;-i) ^ (*o - 00\ “ *V»)

>«<+!

for* > f 0 ,

for £ < fn.

ence now (61:14), (61:15) imply (61:16), (61:17). Consequently (61:14),

562

GENERAL NON-ZERO-SUM GAMES

(61:15) too are necessary and sufficient. Combining (61:14), (61:15) with
part of (61:18), (61:19) we may also write:

Each one of

u t -t 0 - U a -t -i, Vt 9 - V tQ - 1
is greater than each one of

According to the usual ideas, the maximizing t = is the number of
units actually transferred. We have shown that it is characterized by
(61:21), and the reader will verify that (61:21) is precisely Bohm-Bawerk’s
definition of the “ marginal pairs.” 2
So we see:

(61 :A) The size of the transaction, i.e. the number U of units trans-

ferred, is determined in accord with Bohm-Bawerk’s criterion
of the “ marginal pairs.”

To this extent we may say that the ordinary common-sense result has
been reproduced by our theory.

It may be noted, to conclude, that the case when this game is inessential
has a simple meaning. Inessentiality means here

v ((l,2)) = v((l)) + v((2)),

i.e. by (61:9) equality in (61:11). Considering (61:9), (61:10) this means
that the maximum in the latter is assumed at t = 0, i.e. that t 0 = 0. So
we see:

(61 :B) Our game is inessential if and only if no transfers take place

in it — i e. when to = 0. 3

61.6. The Price. Discussion

61 . 6 . 1 . Let us now pass to the determination of the price in this set-up.
In order to provide an interpretation in this respect we must consider more
closely the (unique) solution of our game, as provided by the considerations
of 60.2.2.

Mathematically the present set-up is no more general than the earlier
one analyzed in 61.2.-61.4. : both represent essential general two-person
games, and we know that there exists only one such game. Nevertheless,
that set-up was only a special case of our present one: Corresponding to
s = l. This difference will be felt as we now pass to the interpretation.

1 Comparing the first term of the first line with the second term of the second line is
(61:14); comparing similarly second and first is (61 :15). Comparing first and first is an
inequality from (61:18); second and second, one from (61:19).

1 E. von Bohm-Bawerk: Positive Theorie des Kapitals, 4th Edit. Jena 1921, p. 266ff.

* Note that in our earlier arrangement of 61.2.2. we forced the occurrence of a transfer
by requiring (61:1). Our present set-up leaves both possibilities open.

(61:21)

563

ECONOMIC INTERPRETATION FOR n = 1,2

Comparison of (61:5), (61:6) in (61.3.3.) with (61:9), (61:10) in 61.5.2.
shows that the mathematical identity of these two setups rests upon
substituting the w, v of the former according to

(61:22) u = u ty v = Max*. 0 ,i « (t*,_< + v t ).

The (unique) solution consists, therefore, of all imputations

« = {{<*!, <* 2 j 1

fulfilling (61:3), (61:4) in 61.3.3. In terms of c* 2 this means

(61:23) 0 a 2 - u. 1

Let us now formulate this in terms of the ordinary concept of prices —
instead of the imputations which are the means of expression of our theory. 2
Since, as we concluded in 61.5.3., to units will have been transferred to the
buyer 2, there must be

(61:24) v 1q - t 0 p = a 2 ,

if the price paid was p per unit. Consequently (61:23) means, in terms of
p, that

(61:25) “ u -0 = P = Y 0 Vt o' 8

This can also be written as

<0 *0
(61:26) y 0 ^ ( M — -M “ u — ' ) - v - T 0 ^ ( Vi ~

t-l i-1

61 . 6 . 2 . Now the limits in (61 :26) are not at all those which the Bohm-
Bawerk theory provides. According to that theory, the price must lie
between the utilities of the two marginal pairs named in (61:21) of 61.5.3.,
i.e. in the interval

(61:27)

U »-t 0 + 1 U »-to
> 0+1 ” V *.

This can also be written as

^ p ^

U »-t 0 Ua-t - 1 .

Vt “ V t i
0 0

(61:28) Max (w.-^+i - v< 0+ i - v,) ^ p

^ Min (u.-, - u.-t i, v, - v t i)
0 0 0 0 '

In order to compare this interval with (61:26), it is convenient* to form
a further interval

(61 :29) v, a —t o + 1 u a — t o p =5 Vt 0 i*

1 We could base our discussion equally well on on, but the present procedure is better
suited to be repeated in the case of the three-person market.

* It may be worth re-emphasizing: This is interpretation, and not the theory itself!

* Note that by (61:22) u - u„ v = u,_< f -f t>< 0 .

564

GENERAL NON-ZERO-SUM GAMES

The two last inequalities of (61:20) in 61.5.3. (with t = 0) yield that the
lower limit of (61 :29) is ^ that of (61 :26), and that the upper limit of (61 :29)
is S that of (61 :26). Hence the interval (61 :29) is contained in the interval
(61 :26). Again, (61 :29) obviously contains the interval (61 :27), i.e. (61 :28).
Summing up: The intervals (61:26), (61:29), (61:28) contain each other, in
this order.

So we see:

(61 :C) The price p per unit is limited to the interval (61:26) only,

while Bohm-Bawerk’s theory restricts it to the narrower interval
(61:28).

61.6.3. The two results (61 :A) and (61 :C) give a precise picture of the
relation of our theory, in the present application, to the ordinary common
sense standpoint. 1 They show that there is complete agreement con-
cerning what will happen in fact — i.e. the number of units transferred — but
a divergence as to the conditions under which it will take place — i.e. the
price per unit. Specifically, our theory provided a wider interval for that
price than the ordinary viewpoint.

That the divergence should come at this point and in this direction is
readily understandable. Our theory is essentially dependent upon assum-
ing (among other things) a complete mechanism of compensations among
the players. This amounts to possible payments of varying premiums or
rebates in connection with the various units transferred. Now the narrow
price interval of the ordinary standpoint (defined by Bohm-Bawerk's
“marginal pairs”) is notoriously dependent upon the existence of a unique
price — equally valid for all transfers which occur. Since we are actually
allowing premiums and rebates, as indicated above, the unique price is
obliterated. Our price p per unit is merely an average price — indeed it
was defined as such by (61:24) in 61.6.1. — and it is therefore quite natural
that we obtained a wider interval than the one defined by “marginal
pairs.”

To conclude, we observe that such abnormalities in the formation of the
price structure are also quite in agreement with the fact that the market
under consideration is a bilaterally monopolistic one.

62. Economic Interpretation of the Results for n - 3 : Special Case

62.1. The Case n » 3, Special Case. The Three-person Market

62.1.1. Consider finally n = 3. We propose to obtain an interpretation
in the same sense as was outlined in 61.2.1. This will be done by extending
the model of 61.2.2. dealing with two persons in a market to one dealing
with three persons.

1 We took Bohm-Bawerk’s treatment as representative for that standpoint. Indeed,
the views of most other writers on this subject since Carl Menger are essentially the same
as his.

ECONOMIC INTERPRETATION FOR n - 3: SPECIAL 565

As we have pointed out before, the first mentioned discussion could not
fail to be exhaustive, since there exists only one essential general two-person
game. On the other hand we know that the essential general three-person
games form a 3 parameter family and their detailed discussion in 60.3.2.
forced us to distinguish numerous alternatives. 1 Accordingly several
models would be required to account for all possibilities of the essential
general three-person game. We shall restrict ourselves to the discussion
of one typical class. An exhaustive discussion would be somewhat lengthy
and would not contribute proportionally to our understanding of the theory
— but it would not present any additional difficulties.

62.1.2. We consider accordingly the situation of three-persons in a
market, one seller and two buyers. The discussion of two sellers and one
buyer would lead to the same mathematical setup and to corresponding
conclusions. For the sake of definiteness we discuss the first form of the
problem and leave it to the reader to carry out the parallel discussion of
the second form.

The three participants are 1,2,3 — the seller 1, the (prospective) buyers

2.3. We shall consider successively the special arrangement of 61.2.2.
and the more general one of 61.5.2. In contrast to what we found there, the
latter will now provide a real generalization of the former.

Let us begin with the setup of 61.2.2.: The transaction which we con-
sider is the sale of one (indivisible) unit A of a certain commodity by 1 to
either 2 or 3. Denote the value of the possession of A for 1 by u , for 2 by v,
and for 3 by w.

In order that these transactions should make sense for all participants,
the value of A for each buyer must exceed that for the seller. Also, unless
the two buyers 2,3 happen to be in exactly equal positions, one of them
must be stronger than the other — i.e. able to derive a greater utility from
the possession of A. We may assume that in this case the stronger buyer
is 3. These assumptions mean that we have

(62:1) u < v ^ w.

As in 61.2.2. and 61.5.2. we use for each buyer his original position as the zero
of his utility.

As in 61.5., there is no need to repeat the considerations of 61.2.2.,

61.3. concerning the rules of the game which models this setup.

It is easy to see what its characteristic function must be: Since each
buyer can block sales to him, and the seller as well as both buyers together
can block all sales (cf. 61.5.2.), it follows as in 61.3.3. that

(62:2) v((l)) = u, v((2)) = v((3)) = 0,

(62:3) v((l,2)) = t>, v((l,3)) = w, v((2,3)) = 0,

(62:4) v((l,2,3)) = w. 2

1 The two main cases (a) and (b), the latter being subdivided into the four subcases
represented by the Figures 92-95.

* Of course this makes use of u < v £ w.

566

GENERAL N ON -ZERO-SUM GAMES

This v(S) is a characteristic function, hence it must fulfill the inequalities
of (57:2:a), (57:2:c) in 57.2.1. The verification can be carried out with
little trouble, and is left to the reader.

By the nature of things the game to which v(£) belongs is not constant
sum, 1 hence it is a fortiori essential.

62.2. Preliminary Discussion

62 . 2 . We can now apply the results obtained in 60.3. concerning the
essential general three-person game, to obtain all solutions for our present
problem. We shall again compare the mathematical result with what the
application of ordinary common sense methods gives.

The agreement will turn out to be better than in 61.5.2.-61.6.3. up to a
certain point — specifically the limits to be derived for the price will be the
same with both methods. This is probably ascribable to the fact that we
are dealing now with one unit only, just as in 61.2.2. When we pass to s
units, in 63.1.-63.6., the complications of 61.5.2.-61.6.3. will reappear.

Beyond the point referred to, however, there will be a qualitative
discrepancy between our theory and the ordinary view point. It will be
Been that this is due to the possibility of forming coalitions. This possibility
becomes a reality for the first time for three participants, and it must be
expected that our theory will do it full justice — while the ordinary approach
usually neglects it. Thus the divergence of the two procedures will also
turn out to be a legitimate one from the point of view of our theory.

62.3. The Solutions : First Subcase

62 . 3 . 1 . We proceed to the application of 60.3.1., 60.3.2. to the v(£) of
(62:2)- (62 :4) above.

The imputations in this setup are the

— ►

a = { {«!, a 2 , a 3 | |

with

(62:5) c*i ^ u y ai 0, a 3 ^ 0,

(62:6) + a* + a$ = w .

In order to apply 60.3.1., 60.3.2., it is necessary to bring this to its
reduced form, and then to normalize 7 = 1.

The first operation corresponds to the replacement of our a h a*, a 8 by
the oq, CK 2 , ot^ of

(62:7) a k = a k + a° k

sis mentioned in 57.5.1. and discussed in 31.3.2. and in 42.4.2. The «?>
a obtain as described in the discussion which leads to (59: A) in 59.2.1.

1 Proof: (57:20) in 57.5.2. is violated, e.g. by

v((l)) + v((2,3» - u < la - v((l,2,3)).

ECONOMIC INTERPRETATION FOR n - 3: SPECIAL 567

Specifically,

/AO.QN / W + 2U

(62:8) «i = ol\ s — >

« 2 = «2

w — u

a z = a 8

w — u

The corresponding changes on v(<S) are given by (59:1) in 59.2.1. ; they carry
(62:2)-(62 :4) into

(62:9)

v'((l)) = v'((2)) = v'((3)) = - V-r*,

(62:10) v'((l,2)) =

(62:11)

Zv — 2w — u

v'«l,3)) -

'((2,3)) - -

v'((l,2,3)) = 0.

W — 'll

Thus y = — « — > and so the second operation consists of dividing every-
o

thing by this quantity. Instead of doing this, we prefer to apply 60.3.1.,
60.3.2. directly, inserting everywhere (where 7 = 1 was assumed) the

proportionality factor u • 1

Comparison with (60:8) in 60.3.1. shows that

2 (w — u)
a i — 5 )

w — u

a 2 = o ’

a 3 =

3v — 2w — u

The six lines of (60:15) in 60.3.2., which describe the triangle from which
we derived our solutions, become now:

(62:12)

« i =

w — u

, 2 (w — u)

«1 = Q >

«2 = “
t

<*2 = -

W — U

3 ,

w — u

w — u

3v -- 2w — u

.3

62.3.2. We can now discuss this configuration in the sense of 60.3.3.
Clearly

cti -b 02 “t~ U 3 = v — w ^ 0 ,

hence we have (60:17:b) loc. cit. — i.e. we have the Case (b) id., and it
remains to decide which one of its four subcases, represented by Fig-

1 The procedure is analogous to that used in the discussion of the essential zero-sum
three-person game with excess in 47., in particular in 47.2.2. and 47.3.2. (Case (III)),
47.4.2. (a certain phase of Case (IV)).

* The —1 in (60:15) loc. cit. stands for — 7 , so we must multiply it by the propor-
w — u

tionality factor — g — mentioned above.

*The —ai, — at, —ag in (60:15) loc. cit., which reappear here, include already the
factor — - — •

568 GENERAL NON-ZERO-SUM GAMES

ures 92-95, is present. Therefore we proceed from here on by graphical
representation.

For this representation we use, as before, the plane of Figure 52. Repre-
senting the six lines of (62:12) as those of (60:15) in 60.3.2. were represented

2 (w - u)

by Figures 92-95, we obtain Figure 96. The qualitative features of this
figure follow from the following considerations:

(62:A:a) The second aj-line goes through the intersection of the

first <* 2 - and aj-lines. Indeed:

2(w — u) w — u w ~ u _ n
3 3 3 U ‘

(62:A:b) The two amines are identical.

(62:A:c) The second aa-line is to the left of the first one. Indeed:

It has a greater a' 3 -value, since

3v — 2w — u . w — u . ~

3 + — = w ~ ^ O’

Comparison of this figure with Figures 92-95 shows that it is a (rotated
and) degenerate form of Figure 94 \ l The area ® is degenerated to a point
(the upper vertex of the fundamental triangle A), the areas ®, ® also

1 For this and the remarks which follow, cf. footnote 4 on p. 553.

ECONOMIC INTERPRETATION FOR n « 3: SPECIAL 569

degenerated, but to two linear intervals (the upper and the lower part
of the left side of the fundamental triangle A), while the areas ®, ( 2 ) are
still undegenerated (the trapezon and the smaller triangle, into which the
fundamental triangle A is divided on our figure). This disposition of
the five areas of Figure 94 is shown in Figure 97. The general solution V
now obtains, as stated at the end of 60.3.3., by fitting the picture of Figure 86
into the situation described by Figure 97. Figure 98 shows the result. 1

62.4. The Solutions : General Form

62.4. Before we go any further we note that Figure 97 is of general
validity, assuming

(62:13) u < v £ w,

but the picture it gives refers qualitatively to
(62:14) v < w.

When

(62:15) v = w,

then the area © in Figure 97 — i.e. the upper interval on the left side of the
fundamental triangle — degenerates to a point. (Cf. (62:A:c) in 62.3.2.)
Hence in this case Figure 98 assumes the appearance of Figure 99.

players 2,3 — the two buyers — by:

Assuming (62:14) or (62:15), we may replace (62:13) by the weaker
condition

(62:16) u < v, w.

Let us therefore assume (62:16) only — and not (62:13) with (62:14), (62:15).
This means that each buyer derives a higher utility from the possession of A
than the seller, but it does not place the buyers with respect to each other.
(Cf. the discussion in the first part of 62.1.2.)

1 The curve in Figure 98 is like those in Figure 86, subject to the restriction stated
there: (47:6) in 47.5.5.

570 GENERAL NON-ZERO-SUM GAMES

Now (62:16) leaves three possibilities open: (62:14), (62:15), and
(62:17) v > w.

The solutions of (62:14), (62:15) were given by Figures 98, 99. (62:17)

obtains from (62:14) by interchanging the two players 2,3 — the two buyers
— and v , w. This means that Figure 98 must be reflected on its vertical
middle line (after interchanging v y w). This is shown in Figure 100.
Summing up:

(62 :B) Assuming (62:16), the general solution V is given by Figures

98, 99, 100 for v <, =, > w, respectively.

62.5. Algebraical Form of the Result

62 . 5 . 1 . The result expressed by Figure 98 can be stated algebraically
as follows: 1

The solution V consists of the upper part of the left side of the funda-
mental triangle, and the curve

The first part of V is characterized by

, w — u 3v — 2w — u . , w — u

aj § = “ 3 = 3

Owing to (62:8) in 62.3.1., this means that

c* 2 = 0, w — v a* ^ 0.

Now (62:6) in 62.3.1. gives

a\ = w — a 8 ,

hence the above condition can be written as

(62:18) v ^ ^ w, a 2 = 0, a 8 = w — on.

The second part of V (the curve) extends from the smallest a[ above

to the absolute minimum of a[

Its geometrical shape (cf.

(47:6) in 47.5.5.) can be characterized by stating that along it a 2) a’ z are
both monotonic decreasing functions of a[. We may again pass from
a[, a 2 , a-i to a 2 , a 3 by (62:8) in 62.3.1. Then ai varies from its minimum
in (62:18) above ( v ) to its absolute minimum ( u) } and a 2 , az are again both
monotone decreasing functions of «i. So we have:

(62:19) u g «i ^ v f <* 2 , ctz are monotonic decreasing functions of ai. 2 -*

Thus the general solution V is the sum of the two sets given by (62:18) and

1 Note that it holds whenever v £ w, (62 :B) notwithstanding.

1 They must, of course, fulfill (62:5), (62:6) in 62.3.1.

* As Figure 98 shows, the lowest point on the line / coincides with the highest point on
the curve. I.e. the point ai *= t; of (62:18) and of (62:19) is the same.

Hence we could exclude a\ — v from either (but not from both!) of (62:18), (62:19).

ECONOMIC INTERPRETATION FOR n - 3: SPECIAL 571

(62:19). It will be noted that the functions mentioned in (62:19) are
arbitrary (within certain limits), but that a definite solution (i.e. a definite
standard of behavior) corresponds to a definite choice of these functions.
This situation is entirely similar to those analyzed in (47 :A) of 47.8.2.
and in 55.124.

62.5.2. (62:18), (62:19) can be used whenever v ^ w (cf. footnote 1 on
p. 570). For v = w (62:18) simplifies to

(62:20) ct\ — v, a% = otz = 0.

We shall therefore use (62:18), (62:19) only when v < w, and (62:20),
(62:19) when v = w. 1

If v > w, then we can utilize (62:18), (62:19) by interchanging the
players 2,3 — the two buyers — and v , w . Then (62:18), (62:19) become

(62:21) w 2* ai ^ v, — v — cm, ctz — 0. 2

(62:23) u ^ ai ^ w } a 2 , «3 are monotonic decreasing functions of <*i.*
Summing up:

(62 :C) Assuming (62:16), the general solution is given by (62:18),

(62:19); (62:20), (62:19); (62:21), (62:23) for v <, = , > w
respectively.

62.6. Discussion

62.6.1. Let us now apply the ordinary, common-sense analysis to the
market of one seller and two buyers and one indivisible unit of a good, in
order to compare its result with the mathematical one stated in (62 :C).

The lines of this common-sense procedure are clearly laid down* we are
actually dealing here with one of the simplest special cases of the theory of
“marginal pairs.” The argument runs as follows:

The seller offers only one indivisible unit of the good under consider-
ation and there are two buyers. Hence one will be included in the trans-
action, and one will be excluded. Clearly the stronger buyer will be in
the first position — except when the two buyers happen to be equally strong,
in which case either is eligible. Accordingly the price at which the trans-
action takes place will lie between the limits of the included and the excluded
buyer — and if they happen to be equally strong, the price must be precisely
their common limit. The limit of the seller, which must be assumed to be

l The observation of footnote 3 on p. 570 concerning (62:18), (62:19) applies also to
(62:20), (62:19). Hence we could omit (62:20) altogether, but it is more convenient to
keep it, for the sake of the interpretation in 62.6.

* Note that, owing to the above interchange, (62:4) in 62.1.2. becomes

(62:22) v((l,2,3)) * t>,

and so (62:6) in 62.3.1. becomes

(62:6*) cl i -f* <*2 "I - “3 5=5 v.

* The observation of footnote 3 on p. 570 concerning (62:18), (62:19) applies to (62:21),
(62:23) also. Of course we must replace its v by w.

572

GENERAL NON-ZERO-SUM GAMES

lower than that of either buyer in order to have a genuine three-person
market, comes in no case into play.

In our mathematical formulation the limits of the seller and of the two
buyers were w, v , w. The above remark means

(62:16) u < v, w.

The statements concerning the price amount to

(62:24)

IIA

•S3

IIA

8

for

v < w y

(62:25)

p — V

for

v = w,

(62:26)

w ^ p ^ V

for

v > w.

A buyer who is excluded, finishes where he started — i.e. in our normalization
of utility at zero.

Consequently our present statements correspond exactly to (62:18),
(62:20), (62:21), as provided for by (62:C).

So far the mathematical and the common-sense results agree. But the
limit of this agreement is also in evidence: (62 :C) provided for the further
imputations of (62:19), (62 :23), and there is no trace of these in the ordinary
treatment, as presented above.

What then is the meaning of (62:19), (62:23)? Do they express a
conflict between our theory and the common-sense standpoint?

It is easy to answer these questions, and to see that there exists no real
conflict, but that (62:19), (62:23) represent a perfectly proper extension
of the common-sense standpoint.

62 . 6 . 2 . The amount obtained by the seller in a given imputation, on, is
clearly the price p envisaged when that imputation is offered. In (62:19),
(62:23), ai varies from u to v or w (according to which is smaller) — i.e.
the price varies from the seller’s limit to the weaker buyer’s limit. There
is also a definite (monotonic) functional connection between the (variable)
amounts obtained by the two buyers. 1

These two facts strongly suggest giving (62:19), (62:23) the following
verbal interpretation: The two buyers have formed a coalition, based on a
definite rule of division for any profit obtained, and are bargaining with the
seller. The rule of division is embodied in the monotonic functions that
occur in (62:19), (62:23). No bargaining can depress the seller under his
own limit. 2 On the other hand a price above the limit of the weaker buyer
would exclude him from any possibility of exerting influence.

The specific rules contained in (62:19), (62:23), and the roles of all
participants in these situations may be given more extended verbal treat-
ment. We shall not do this here, since the above should suffice to establish
our main point: On the one hand (62:18), (62:20), (62:21) (i.e. the upper
parts of V in Figures 98-100) correspond to the competition of the two
buyers for the transaction — in which the stronger player, if one exists, is

1 All these “amounts” refer to the utility in which we reckon — of the goods under
consideration there exists only one, indivisible, unit.

1 The seller's limit is his best alternative use (instead of a sale) for A.

ECONOMIC INTERPRETATION FOR n = 3: GENERAL 573

sure to win. On the other hand, (62:19), (62:23) (i.e. the lower part of V
in Figures 98-100, the curves) correspond to a coalition of the two buyers,
against the seller.

Thus it appears that the classical argument — at least in the form used
in 62.6.1. — gives the first possibility only, disregarding coalitions. Our
theory, to which the coalitions contributed decisively from the beginning, is
necessarily different in this respect: It embraces both possibilities, indeed
it gives them welded together, as a unit, in the solutions which it produces.
The separation, according to schemes with and without coalitions, appears
only as a verbal comment on the relatively simple three-person game —
there is no reason to believe that it can be carried out for all games, while
the mathematical theory applies rigorously in all situations.

63 . Economic Interpretation of the Results for n = 3 : General Case

63.1. Divisible Goods

63 . 1 . 1 . It remains for us to extend the three-person setup of 62.1.2. in
the same way that the two-person setup of 61.2.2. was extended in 61.5.2.,
61.5.3.

Let us accordingly return to the situation described in 62.1.2. : the seller 1
and the (prospective) buyers 2, 3 in a market. We allow now for trans-
actions involving any or all of s (indivisible and mutually substitutable)
units A i 9 • • • , A, of a particular good. (Cf. also footnote 1 on p. 560.)
Denote the value of t (= 0, 1 , • • ■ , s) of these units by u t for 1 , by v t
for 2, and by w t for 3. Thus the quantities

(63:1) u 0 = 0, iii, • • • , u„

(63:2) v 0 = 0, i>i, • • • , v„

(63:3) wo = 0, w h • • • , w„

describe the variable utilities of these units for each participant.

As before, we use for each buyer his original position as the zero of his
utility.

As in 61.5.2., 61.5.3. and 62.1.2., we need not repeat the considerations
of 61.2.2., 61.3.1., 61.3.2. concerning the rules of the game which models
this setup.

It is easy to see what its characteristic function must be: Since each
buyer can block sales to him, and the seller as well as both buyers together
can block all sales (cf. 61.5.2. and 62.1.2.), it follows as in 61.3.3. that

(63:4) v((l)) = v((2)) = v((3)) - 0,

(63:5) v((2,3)) = 0.

Denoting the number of units transferred from the seller 1 to the buyers 2,
3 by t , r, respectively, it is easy to express what the remaining coalitions
(1,2), (1,3), (1,2,3) — i.e. the seller with either one or both buyers — can

574

GENERAL NON-ZERO-SUM GAMES

achieve. The familiar arguments give

v((l,2)) = Max(_ 0 ,i , ( u + v t ),

( ) v((l,3)) = Max r _ 0 ,i (w._ r + »,),

(63:7) v((l,2,3)) = Max ( , r _o,i , (w,_i_ r + Vt + w r ). 1

<+r5a >

This v(S) is a characteristic function. We leave it to the reader to verify
the inequalities which are implied thereby.

The discussion as to when this game is essential can be carried out as in
61.5.2., 61.5.3. and is left to the reader. 2 It is also possible to determine
when one of the two buyers 2,3 becomes a dummy in the sense of our theory
of decomposition. We shall not consider this either; the result is not
difficult to obtain, and though not surprising it is not uninteresting.

63*1.2. Restricting r in the maximum of (63:7) to the value 0 converts
this into the first maximum of (63:6). Restricting t there to the value 0
converts it into u t . By each one of these operations the value becomes ^ ,
i.e. we have

(63:8) v((l)) ^ v((l,2)) ^ v((l,2,3)).

If we do the same to r, t in the reverse order, we obtain similarly
(63:9) v((l)) g v((l,3)) g v((l,2,3)).

Consider the first inequality in (63:8). To have equality there means
that the first maximum in (63:6) is assumed for t = 0. According to
the usual ideas on the subject this means that the seller and buyer 2, in the
absence of buyer 3, would effect no transfers. I.e., that buyer 2, in the
absence of buyer 3, is unable to make the market function.

Consider the second inequality in (63:8). To have equality there
means that the maximum in (63:7) is assumed for r = 0. According to the
usual ideas on the subject, this means that the seller and buyer 3, in the
presence of buyer 2, would effect no transfers. I.e., that buyer 3, in
the presence of buyer 2, is unable to participate in the market.

Summing these up, together with the corresponding statements for
(63:9) which obtain by interchanging buyers 2, 3, we have:

(63: A) Equality in any one of the four inequalities of (63:8), (63:9)

is a sign of some weakness of one of the buyers.

In the first inequality of (63:8), [(63:9)] it means that
buyer 2 [3], in the absence of buyer 3 [2], is unable to make the
market function. In the second inequality of (63:8), [(63:9)] it
means, that buyer 3 [2], in the presence of buyer 2 [3], is unable
to influence the market.

1 The extra condition t -f r ^ 8 under this Max expresses that the number t -4- r of
units sold cannot exceed the number of units originally possessed by the seller.

1 The discussion of the relationship with (62:1) in 62.1.2. or with (62:16) in 62.4.,
when 8 — 1, can also be carried out easily. The discussion of (61 :B) at the end of 61.5.3.
and footnote 3 on p. 562 should be remembered.

ECONOMIC INTERPRETATION FOR n - 3: GENERAL 575

The really interesting case arises, obviously, when all these weaknesses
are excluded. It is therefore reasonable to make the hypotheses which
express this:

(63:B:a) We have < in the first inequalities of both (63:8) and

(63:9).

(63:B:b) We have < in the second inequalities of both (63:8) and

(63:9).

63.2. Analysis of the Inequalities

63 . 2 . 1 . Assume for a moment (63:B:a), but the negation of (63:B:b).
This means that one of the two players is absolutely stronger than the other.
More precisely: That he is at least as strong as the other player, even when
he tries to exclude the other player completely from the market.

Hence we may expect in this case a result which is similar to that
obtained in 62.1.2.-62.5.2., when only one (indivisible) unit A was available.
I.e. the divisibility of the supply into units Ai, • • • , A $ which we have
here, should now become effective.

This is indeed the case. To prove it, introduce the quantities u , v, w by

(63:10) v((l)) = u , v((l,2)) = v, v((l,3)) = w.

Then the second inequality of (63:8) and of (63:9) and the negation of
(63:B:b) give

(63:11) v((l,2,3)) = Max (v } w)

while the first inequality of (63:8) and of (63:9) and (63:B:a) give

(63:12) u < v, w.

Now we have precisely the conditions of 62.1.2.-62.5.2.: (63:12) coincides
with (62:16) in 62.4., while (63:4), (63:10) give (62:2), (62:3) in 62.1.2., and
(63:11) gives (62:4) in 62.1.2. (when v ^ w) or (62:22) in 62.5.2. (when
v £ w).

Consequently the results of 62.4. and 62.5.2. are valid, with the u, v, w
of (63:10). The general solution obtains as described, e.g. in (62 :B) in
62.4., according to Figures 98-100.

63 . 2 . 2 . From now on we assume that (63:B:a), (63:B:b) are both valid.
We introduce the quantities u, v , w , z by

(63:13) v((l)) = u, v((l,2)) = v, v((l,3» = w,

(63:14) v((l,2,3)) = *.

Then (63:8), (63:9) and (63:B:a), (63:B:b) state that

u <

< 2 .

(63:15)

v

w

576

GENERAL NON-ZERO-SUM GAMES

This arrangement differs from that of 62.1.2., but it is nevertheless
worth while to compare them in detail: (63:15) corresponds to (62:1)
loc. cit., and (63:4), (63:13), (63:14) correspond to (62:2)-(62:4) id.

It is now convenient to introduce again the assumption of decreasing
utility , already utilized in 61.5.2., 61.5.3. In fact we need it now at a
somewhat earlier stage than we did then : It is now (at least in part) useful
in the mathematical part of the theory, 1 while we needed it there only in
the interpretative part.

We state the decrease of utility for all three participants 1,2,3,:

(63:16) Ui — uo > u% — u x > • * • > u, — u a - 1 ,

(63:17) Vi — t; 0 > v* — Vi > • • • > v, — v 9 -i,

(63:18) Wi — w 0 > w 2 — Wi > • • • > w, — i.

In the immediate application only (63:16) will be required. This is it:

(63:19) v + w > z + u}

Proof: Owing to (63:6), (63:7) and (63:13), (63:14), the assertion (63:19)
can be written as follows :

Maxi_ 0 , i (w— t + v t ) + Max r _o, i (w.~r + w r )

> Max*, r _o. 1 + Vt + Wr) + u 9 .

t+r£»

Consider the t, r for which the maximum on the right-hand side is assumed.
Since we have (63:B:b), i.e. < in the second inequalities of (63:8), (63:9),
we can conclude from the argumentation of 63.1.2. that these t } r are ^ 0.
We denote them by to, r 0 . Hence our assertion is this

Max<« 0t i • (u,-t + v t ) + Max r _ 0t i « (u«- r + w r )

> U,- t -r 0 + V to + + U 9 .

I.e., we claim: There exist two t , r with

W,_| + V t + W,_ r + W r > U$ — — r Q + Vt 0 + Wr o + U 9 .

Now this is actually the case for t = to, r = r 0 . The above inequality
may then be written as

(63 :20) u§—r 9 ^•—* 0 ”* r 0 ^

It should be conceptually clear that this follows from our assumption of
decreasing utilities. Formally it obtains from (63:16) in this way: (63:20)
states that

1 But not necessary; the absence of this property would only complicate the discussion
somewhat.

* In 62.1.2. this was trivially true. Indeed using (63:13), (63:14) we obtain in that
case

U < V ^ w « z

and this gives (63:19) immediately.

ECONOMIC INTERPRETATION FOR n = 3: GENERAL 577

*0

(63:21) 2 («_, _t+l > 2) (w,_i+l ~ M«-i).

»-l i-1

(63:16) implies

W*' W|'— i ^ W|" W| w -i

whenever $' < hence in particular

i+l r 0 — < ^ 'Wf—i+l Ut—tf

and from this (63:21) follows.

63.3. Preliminary Discussion

63 . 3 . We now apply 60.3.1., 60.3.2. to the present setup. This will
prove to be quite similar to the application carried out in 62.3. for the setup
of 62.1.2. The exposition which follows will therefore be more concise, and
is best read parallel with the corresponding parts of 62.3.

As to the comparison of the mathematical result with that of the ordi-
nary, common sense approach, the remarks of 62.2. apply again. We
indicated already there what complications the present setup produces.
We shall consider the situation only briefly, although it is a rather important
one. The general viewpoints were sufficiently illustrated by our earlier,
simpler examples, and the specific, detailed interpretative analysis of this
setup — and other, even more general ones — will be taken up suo jure in a
subsequent publication.

63.4. The Solutions

63 . 4 . 1 . The imputations in the present setup are the

a = {«i, a 2 , <*s}

with

(63:22) a\ ^ u y a* 0, as ^ 0,

(63:23) ai + ot% + as = 3-

It is again necessary to introduce the reduced form. This amounts to a
transformation

(63:24)

a£ = <*k + a£

as described in 62.3. The processes discussed there determine the aj, aj,
ajj, so that (63:24) now becomes

(63:25)

, z + 2u

<*i = ai g — >

aj= a 8 —

Z — U

3

The corresponding changes on v(S) are again given by (59:1) in 59.2.1.;
they carry (63:4), (63:13), (63:14) into

578

GENERAL NON-ZERO-SUM GAMES

(63:26) v'((l)) = v'((2)) = v'((3)) =

(63:27) v'((l,2)) = — ~ v'((l,3))

(63:28) v'((l,2,3)) = 0.

2 — U

~3 '

Zw — 2z — u
3 ’

v'((2,3)) = - 2(Z f — } »

Z XL • • •

Thus 7 = —= — ♦ and we again refrain from passing to the normalization

o

y = l.

Hence we must again insert a proportionality factor when applying
60.3.1., 60.3.2. as described in 62.3. This proportionality factor is now
z — u

Comparison with (60:8) in 60.3.1. shows that now

2(z — u) 3w — 2z — u

ai== 3 —' a2== 3 '

a 8 =

3v — 2z — u

The six lines of (60:15) in 60.3.2. which describe the triangles from which we
derived our solutions, become now:

«i - “

(63:29) < =

3

2 (z — u)

«2 = -

z — u

«8 = ”

z — u

a 2 =

3w — 2z — u
3 '

= ”

3v — 2z — u

63.4.2. Applying the criterium of 60.3.3., we find that
cli + a 2 + a 8 = v + w — 2z g 0.

Hence we have again (60:17 :b) loc. cit. — i.e. the Case (b) id., and it remains
to be decided which one of its four subcases, represented by Figures 92-95,
is present.

Following the same procedure of graphical representation as in 62.3.,
we obtain Figure 101. The qualitative features of this figure follow from
the following considerations :

(63:C:a) The second cq-line goes through the intersection of the first

<*2- and aj-lines. Indeed:

2(z — u) z — u z — u _ n
3 3 ~ 3 U *

The second a£- [a,-] line is to the right [left] of the first one.
Indeed: It has a greater as- [a 8 -] value, since

(63 :C :b)
(63:C:c)

ECONOMIC INTERPRETATION FOR n = 3: GENERAL 579

3u> — 2z — u , z — u ^ n

3 + -y = * ~ w > 0.

3v — 2z — u , z — u ^ ~

g 1 g— = Z ~ V > 0.

(63:C:d) The first a' r line lies below the intersection of the second a%

and aj-lines. Indeed:

z — u — 2z — u 3t> — 2 z — u

= z + u — v — w<0.

by (63:19) in 63.2.2.

Comparison of this figure with Figures 92-95 shows that it is again a
(rotated and) degenerate form of Figure 94 (cf. footnote 1 on p. 568),
although less degenerate than the corresponding Figure 96 in 62.3.: The

Figure 103.

area ® is again degenerated to a point (the upper vertex of the fundamental
triangle A), but the areas 0, ®, ® are still undegenerated (the four

areas into which the fundamental triangle is divided in our figure). This
disposition of the five areas of Figure 94 is shown in Figure 102. The
general solution now obtains as stated at the end of 60.3.3., by fitting the
picture of Figure 86 into the Situation described by Figure 102. Figure 103
shows the result (cf. footnote 1 on p. 569).

580

GENERAL NON-ZERO-SUM GAMES

Summing up:

(63 :D) Assuming (63 :B :a), (63 :B :b) and (63:16), the general solution

V is given by Figure 103.

Comparison of this figure with those of 62.3.-4. shows that Figure 103
is a form intermediate between those of Figures 98-100, and those figures
are in turn degenerate forms of Figure 103.

63.6. Algebraic Form of the Result

63 . 5 . The result expressed by Figure 103 can be stated algebraically,
in the same way as was done for Figure 98 in 62.5.1.

In Figure 103 the solution V consists of the area m and the curve ~ .
The first part of V is characterized by

3w — 2z — u
3

^ «2 ^

z — u

3

Owing to (63:25) in 63.4.1., this means that

z — w ^ a 2 ^ 0, z — v ^ a 3 ^ 0.
Now (63:23) in 63.4.1. gives

ot\ — Z — a .2 — a 8 ,
and so the exact range of ai is

v-\-w — Z^ai^Z.

(Recall that v + w — z > u by (63:19) in 63.2.2.) We state all these
conditions together, the result being somewhat more complicated than its
analogue (62:18) in 62.5.1. It is this:

(63:30)

v + w — z^ai^z,
oti + a 2 + a 8 = Z.

0 ^ a 2 g z — w, 0 ^ a s ^ z — v,

The ranges in the first line of (63:30) are the precise ones for ai, a 2 , a 8 .

The second part of V (the curve) can be discussed literally as in 62.5.1. :
ai varies from its minimum in (63:30) above (v + w — z) to its absolute
minimum (u), and a 2 , a 8 are monotonic decreasing functions of a\. So
we have:

(63:31) w g ai ^ v + w — z, a 2 , a 8 are monotonic decreasing functions

ofai. 1,2

Thus the general solution V is the sum of the two sets given by (63 :30)
and (63:31). It will be noted that the role of the functions in (63:31) is
the same as that discussed at the end of 62.5.1.

1 They must, of course, fulfill (63:22), (63:23) in 63.4.1.

* As Figure 103 shows, the lowest point in the area ■ coincides with the highest point
on the curve. I.e. the point at — v -f w — z of (63:30) and of (63:31) is the same.

Hence we could exclude ai *■ v + w — z from either one (but not from both!) of
(63:30), (63:31).

ECONOMIC INTERPRETATION FOR n - 3: GENERAL 581

Summing up:

(63 :E) Assuming (63:B:a), (63:B:b) and (63:16), the general solu-

tion V is given by (63:30), (63:31).

63.6. Discussion

63 . 6 . 1 . Let us now perform the equivalent of 62.6. and apply the ordinary
common-sense analysis to the market of one seller and two buyers and 8
indivisible units of a particular good, in order to compare its result with the
mathematical one stated in (63 :E).

Actually the interpretation which ought to be carried out now must
combine the ideas of 61.5.2.-61.6.3. with those of 62.6.: the former apply
because we have divisibility into $ units; the latter because the market is
one of three persons. As indicated in 63.3., we do not propose to go fully
into all details on this occasion.

The two parts (63:30), (63:31), of which our present solution consists
are closely similar to the two parts (62:18), (62:19) (or (62:20), (62:19), or
(62:21), (62:23)) obtained in 62.5. (Cf. also (63 :E) in 63.5. with (62:C)
in 62.5.2.) It appears most reasonable, therefore, to interpret them in the
same way as we did in the corresponding situation in 62.6.2.: (63:30)
describes the situation where the two buyers compete for the s units in the
seller’s possession, while (63:31) describes the situation where they have
formed a coalition and face the seller united. The reader will have no
difficulty in amplifying the details, in parallel to 62.6.2.

These being accepted, there is nothing new to be said about (63:31), the
situation in which the buyers have combined and do not compete. (63:30)
however, which describes their competition, still deserves some attention.

Let us consider the imputations belonging to (63 :30), and let us formulate
their contents in terms of the ordinary concept of prices. This is the
equivalent of what we did at the corresponding point in 61.6.1., 61.6.2.

We introduce again the t , r for which the maximum in

(63:7) v((l,2,3)) = Max <(r . 0 .i « + v t + w r )

is assumed: t 0f r 0 . Since our imputations

a = { {ai, a 2 , a 3 } } with ai + aj + a 3 = v((l,2,3))

actually distribute the amount v((l,2,3)), these t 0> r<j must represent the
numbers of units actually transferred by the seller to the buyers 2,3,
respectively.

The analysis of 61.5.2., 61.5.3. leading up to (61 :A), could now be
repeated mutatis mutandis. It would show that numbers of units trans-
ferred, f 0 , r 0 , can be described in accord with Bohm-Bawerk’s criterion of
“ marginal pairs” — just as was done loc. cit. for the corresponding number
of transfers t 0 . Since this discussion would bring up nothing new, we shall
not dwell upon the point any further.

582 GENERAL NON-ZERO-SUM GAMES

63.6.2. We now turn to the question of prices. The buyers 2,3 received,

as we saw, to, r<> units, respectively. The imputation a on the other hand,
ascribes them the amounts a*, a 8 . These two descriptions can be harmo-
nized only by establishing the equations

(63:32) Vi # - t 0 p = at,

(63:33) Wr t — r 0 q = aa,

and interpreting p , q as the prices paid per unit by the buyers 2,3 respec-
tively. (63:32), (63:33) are the equivalents of (61:24) in 61.6.1., but it
must be emphasized that we obtained two different prices for the two
buyers!

Now (63:30) can be stated in terms of p, q 1 as follows:

(63:34) (i vt - z±w)£p^^ v t ,

1 0 1 0

(63:35) — (w r — z + v) g q ^ ~ w r .

r o • to •

These inequalities are the analogues of (61:25) in 61.6.1. We could treat
them in a way similar to that there, and compare them with those limits
which result from the application of Bohm-Bawerk’s theory. We shall not
carry this out in detail for the reasons stated in 63.3. A few remarks may
nevertheless be appropriate.

The intervals (63 :34) and (63 :35) are again wider than those of Bohm-
Bawerk’s theory — just as in 61.6. (cf. (61 :C) id.). Some numerical examples
show, however, that the difference tends to be smaller. It is therefore
possible — although nothing has been proven in this respect — that a further
increase in the number of buyers may tend to obliterate this discrepancy
in that part of the solution which corresponds to no coalition between the
buyers. This surmise must, however, be considered with the greatest
caution, since we know too well how rapidly the complication of solutions
increases with the number of participants and how difficult the interpreta-
tion of different parts of the solution may then become.

It will be observed also that we had to introduce two (possibly) different
prices for the two buyers, and this in spite of our, still valid, assumption
of complete information. This is perfectly in harmony with the interpre-
tations of 61.6.3.: We saw there that what we called prices were really only
average prices of several different transactions, that the seller and the
buyers must have been operating with premiums and rebates — and all this
is necessarily conducive to a differentiation between the two buyers.

Finally, we may state the equivalent of the last remark of 61.6.3. All
these abnormalities in the formation of the price structure are also quite in

1 1.e. its statements concerning as, a t can be translated by means of (63:32), (63:33)
into statements on p, q.

The statement of (63:30) concerning ai is merely a consequence of those on as, at
using ai -f as 4- «* ■* *• Hence it need not be considered.

THE GENERAL MARKET 683

agreement with the fact that the market under consideration is one of a
monopoly versus duopoly.

64. The General Market
64.1. Formulation of the Problem

64.1.1. The markets which we have considered so far were very
restricted: They consisted of two or of three participants. We shall now
go a step further and consider a more general market, which consists of
l + m participants: l sellers and m buyers. This is, of course, still not the
most general arrangement: That would have to provide — among other
things — for the possibility that each participant can choose whether he will
buy or sell; or again, that he may be seller for one class of goods, and buyer
for another. In this study, however, we shall content ourselves with the
above case.

Further, we propose to consider one kind of goods only, of which s units
Ai y • • • , A, are available.

It is convenient to denote the sellers by 1, • • • , Z, their set by

L - ( 1 , * • • , 0 ;

the buyers by I *,•••, m*, their set by

M = (1*, • • • ,m*);

and the set of all participants by

/ = LuM = (1, • • • , J, 1*, • • • , m*). 1

Denote the number of units of the goods under consideration, originally
in the possession of the seller i by s t . Then clearly

i

(64:1) Z Si = s -

i-1

Denote the utility of t{ = 0, 1, • • • , s») units of the goods to the seller i by
u\ and the utility of = 0, 1, • • • , s) units of the goods to the buyer j* by
v {*. Thus the quantities

(64:2) Uq == 0, u\ f , u ^ (t ' = 1, , £)>

(64:3) w’o* = 0, i**, • • • , i** (j = 1 V * * > ™*)>

describe the variable utilities of these units to each participant.

As before, we use for each buyer his original position as the zero of his
utility.

As in 61.5.2., 61.5.3., 62.1.2. and 63.2.1., we need not repeat the con-
siderations of 61.2.2., 61.3.1., 61.3.2. concerning the rules of the game which
models this setup.

1 We use this notation instead of the conventional 1, * • * , l, l + b * * * » * + m.

584 GENERAL NON-ZERO-SUM GAMES

64.1.2. The determination of the characteristic function v(<S) of this
game is easy:

Clearly S c / = L u M . We now consider successively three alter-
native possibilities.

First: S sL. In this case S consists of sellers only, who can carry out
no transactions among themselves. One sees immediately that v(S)
merely states their original position :

(64:4) v(S) = X <■

tin 5

Second: S q M. In this case S consists of buyers only, who are equally
unable to carry out any transactions among themselves. One sees again
that v(S) merely states their original position:

(64:5) v(S) = 0.

Third: Neither S c L nor £ c M — i.e. S has elements in common both
with L and with M . In this case S contains sellers as well as buyers, hence
transactions between these are definitely possible. On the basis of these,
the following formula obtains:

(64:6) v(S) = Max^-o, i, • • • , «,(»insriL) f £ u\ % + £

— 0, 1, • • • , •(;* in 5 fl Af ) in S H L j* in S 0 M

2 t, + 2 v “ S

X in S n L j* in S H M tin S Cl L

In this expression S n L is the set of all sellers in S, S n M the set of all
buyers in S , U the number of units transferred from the seller i (in S n L),
r,* the number of units transferred to the buyer j * (in S n M). 1 The reader
will now have no difficulty in verifying the formula (64:6).

64.2. Some Special Properties. Monopoly and Monopsony

64.2.1. We are far from being able to discuss the theory of this game —
the market of l sellers and m buyers — exhaustively. We have at present
only some fragmentary information on special cases and beyond this only a
few surmises concerning wider areas. The problems which arise in this
connection seem to be of definite mathematical interest, aside from their
economic importance. It would seem premature, however, to discuss this
subject before the investigation has penetrated deeper.

Instead we shall draw some immediate conclusions from the two simpler
of our equations: (64:4), (64:5). They are as follows:

(64 :A) All sets S € L and all sets S s M are flat.

1 There is no need to state here which seller is transferring each particular unit to
which buyer: The resulting utilities — which alone enter into v(S) — are not affected by
this.

All negotiations between individuals, coalitions, compensation, etc., must be
automatically taken care of by the application of our theory.

THE GENERAL MARKET

585

Proof: This means that

v(S) = Y, v((k)) for SqL and for S c M .

kin 8

which follows immediately from (64:4), (64:5).

(64 :B) The game is constant sum if and only if it is inessential.

Proof: Sufficiency: Inessentiality clearly implies constant-sum.

Necessity: Assume that the game is constant-sum.

As L f M are complementary sets

(64:7) v(J) = v(L) + v(M).

Now by (64 :A) (with S = L f M)

(64:8) v(L) = £ v((*)), v(M) = £ v((*)).

k in L kin M

Combining (64:7) and (64:8) we obtain
(64:9) v(7) = £ v((*)).

k in I

Now the modification of (27 :B) in 27.4. which applies according to 59.3.1.
in our case, gives just (64:9) as a criterion of inessentiality.

It may be worth noting that the criterion of inessentiality (64 :9) becomes,
when stated explicitly by means of (64:4)-(64:6), this: The maximum in
(64 :6) is equal to £ u\ .. Now this is the value of the expression maximized

tin L

in (64 :6) when U s s », r,* = 0. So the statement becomes, that the maxi-
mum in (64:6) is assumed when U = s t) r ; . = 0, i.e. when no transactions
take place.

Hence (64 :B) can also be formulated as follows:

(64 :B*) The fact that the individual utilities of the sellers and buyers

are such that no transactions take place at all — i.e. that the
maximum in (64:6) is assumed when U = s if r, * == 0 — is
equivalent to these: that the game is constant-sum; or equiva-
lently (in this case!) that it is inessential.

The salient point of this result is that our game, representing a market,
can be constant-sum only at the price of the market being absolutely
ineffective. Hence this problem belongs quite intrinsically to games of
nonconstant-sum.

64 . 2 . 2 . We now continue in a somewhat different direction.

(64 :C) Consider two imputations,

a = {{ai, * • • , on, «!•, • • *

8 = 1 \8l, * • ’ # Ph £l*l * * ' 9 8m* li.

586

GENERAL NON-ZERO-SUM GAMES

Assume that

a H 0,

S being the set of 30.1.1. for this domination. Then neither
SnL nor S n M can be empty. 1

Proof: Otherwise we should have S £ M or S £ L. Hence S is flat by
(64:A) and therefore certainly unnecessary (cf. 59.3.2.).

We conclude from (64 :C) that in this case

(64:10)

(64:11)

oti > P i for at least one i in L,

ay* > Pi* for at least one j* in M .

These formulae (64:10) and (64:11) have a role of some interest, when
either L or M is a one-element set: l = 1 or m = 1. This means that there
exists precisely one seller or precisely one buyer, — i.e. that we have monopoly
or monopsony.

In these cases the i of (64:10) or the,;* of (64:11) is uniquely determined:
i = 1 or j* = 1*. Se we have:

(64 :D) a H P implies

(64:12) ai > pi

(64:13) a, > pi *

i- 1,

m = 1.

The remarkable thing is that both (64:12) and (64:13) are transitive

relations, while domination a h P is not. There is, of course, no con-
tradiction in this, — (64:12) or (64:13) is merely a necessary condition for

a H P . But it is nevertheless the first time that the domination concept
in an actual game is so closely linked to a transitive relation.

This connection seems to be a quite essential feature of the monopolistic
(or monopsonistic) situations. 2 It will play a role of some importance in
65.9.1.

1 I.e. S must contain both sellers and buyers.

•The verbal interpretation of (64:12), (64:13) is simple and plausible: No effective
domination is possible without the monopolist (or monopsonist).

CHAPTER XII

EXTENSIONS OF THE CONCEPTS OF
DOMINATION AND SOLUTION

65. The Extension. Special Cases

65.1. Formulation of the Problem

65.1.1. Our mathematical considerations of the n-person game beginning
with the definitions of 30.1.1. made use of the concepts of imputation,
domination and solution, which were then unambiguously established.
Nevertheless in the subsequent development of the theory there occurred
repeatedly instances where these concepts underwent variations. These
instances were of three kinds:

First : It happened in the course of our mathematical deductions, based
strictly on the original definitions, that concepts rose to importance which
were obviously analogous to the original ones (of imputation, domination,
solution) but not exactly identical with them. In this case it was conven-
ient to designate them by those names, necessarily remembering the differ-
ences. Examples of this are to be found in the investigation of the essential
three-person game with excess in 47.3.-47.7. where the discussion of the
fundamental triangle is reduced to that of one of the various smaller triangles
in it. Another example is offered by the investigation of a special simple
n-person game in 55.2.-55.11., where the discussion of the original domain is
reduced to that one of V' in Ct (cf. the analysis of 55.8.2., 55.8.3.).

Second: In the course of our considerations on decomposability in
Chapter IX, we explicitly re-defined (generalized) the concepts of imputa-
tion, domination and solution in 44.4.2.-44.7.4. This corresponded to an
extension of the theory from zero-sum to constant-sum games. Throughout
what followed we emphasized that we were investigating a new theory,
analogous to, but not identical with, the original one of 30.1.1.

Actually these two types of variations of our concepts are not funda-
mentally different: The second type can be subsumed under the first one.
Indeed, the new theory was introduced in order to handle the problem of
decomposition of the original one more effectively. This motive was
stressed throughout the heuristic considerations which led to this generaliza-
tion. In the analysis of imbedding in 46.10., particularly in (46 :K) and
(46 :L) there, we established rigorously that the new theory can be sub-
ordinated to the original one precisely in this sense.

Third: The concepts of imputation, domination and solutions were again
re-defined (generalized) in Chapter XI, specifically in 56.8., 56.11., 56.12.

587

588

EXTENSIONS OF THE CONCEPTS

This corresponded to the final extension of the theory to general games.
We again emphasized that from there on we were investigating a new theory
analogous to, but not identical with, the preceding ones.

This generalization was, however, fundamentally different from the two
preceding ones: It represented a real conceptual widening of the theory and
not a mere technical convenience.

65 . 1 . 2 . Throughout the changes referred to above it was in evidence
that while the concepts of imputation, domination and solution varied
(particularly regarding extension), some connection among them remained
invariant. In order to acquire a general insight into these changes — and
other analogous ones which may follow — it is necessary to find a precise
formulation of this invariant connection. When this is done we can permit
complete generality in all respects and reformulate the theory on that basis.

By recalling the instances enumerated in 65.1.1., it will appear that this
invariant connection is the process by which the concept of a solution is
derived from those of imputation and domination. This is the condition
(30:5:c) (or the equivalent ones (30 :5 :a) and (30:5:b)) in 30.1.1. Hence
we reach perfect generality if we release the notions of imputation and
domination from all restrictions, but define the solutions in the way
indicated.

In accordance with this program we proceed as follows :

Instead of imputations we consider the elements of an arbitrary but
fixed domain (set) D.

Instead of domination we consider an arbitrary but fixed relation S
between the elements x , y of D. 1

Now a solution (in D for S) is a set V £ D which fulfills the following
condition:

(65:1) The elements of V are precisely those elements y of D for

which x$y holds for no element x of V. 2

65.2. General Remarks

66 . 2 . These definitions provide the basis for a more general theory in the
sense indicated.

It should be noted that our present concept of solution bears the same
relation to that one of saturation analyzed in 30.3. and particularly in 30.3.5.,
as the original concept of 30.1.1. In particular our (65:1) should be com-
pared with the fourth example in 30.3.3., our present S corresponding to the
negation of the (R there. It is especially significant that in the search for
solutions all difficulties connected with the lack of symmetry of the relation
considered, arise again. I.e., the remarks made in 30.3.6. and 30.3.7. to
this effect apply once more.

1 x%y expresses that this relation holds between the specific elements x and y. The
reader should recall the discussions at the beginning of 30.3.2.

* This is the equivalent of (30:5:c) in 30.1. 1., as promised.

THE EXTENSION. SPECIAL CASES 689

We shall see subsequently how these difficulties can be resolved at least
in some specific cases. 1

In order to acquire a better understanding of the entire situation, we
must consider some specializations of the relation x$y. Indeed, in our
present exposition 8 is entirely unrestricted and we cannot expect to find
any particularly deep result while 8 remains in this generality. On the
other hand, the original concept of a solution, as defined in 30.1.1., remains
the most important application of 8 and it seems very difficult to discover
any simple distinguishing properties of this particular relation. Therefore
there is no apparent way to introduce specialization, however desirable this
would be.

We will nevertheless discuss three frequently used schemes of specializa-
tion for relations xSy and finally find a fourth one which possesses a certain
limited applicability to our problem proper. In order to carry this out,
we need a few mathematical preparations which follow.

66.3. Orderings, Transitivity, Acyclicity

66 . 3 . 1 . We first consider such relations xSy (with the domain D) which
share the essential features of the concepts “greater” and “smaller.” This
order of ideas has received detailed and careful considerations in the mathe-
matical literature and there exists today rather general agreement to the
effect that a complete list of these properties runs as follows:

(65:A:a) For any two x, y of D one and only one of the three follow-

ing relations holds :

x = y, x&y, ySx.

(65:A:b) xS y ) ySz together imply xSz . 2

We call a relation 8 with these properties a complete ordering of D.

Examples of complete orderings are easy to give and conform to ordinary
intuition: The usual concept of “greater” for the set of all real numbers or
for any part of it.* The concept of “smaller” under the same conditions.
Even the points of the plane possess complete orderings, e.g. this one: xSy
means that x must have a greater ordinate than y or the same one, but then
x must have a greater abscissa than y. A

66 . 3 . 2 . The concept of complete ordering can be weakened considerably
so that a significant concept still remains. This, too, has received attention
in mathematical literature 5 and is of importance in the theory of utilities.
It obtains by weakening (65:A:a) above, but retaining (65:A:b) unchanged.

1 Cf. the results of 65.4., 65.5., and the less superficial ones of 65.6.-65.7.

f The reader who substitutes the ordinary “ greater ” relation x > y for xSy in
(65:A:a), (65:A:b), will verify that these are indeed the basic properties of “ greater.”

* E.g. the integers, or any interval, etc.

4 Without this last proviso, our S would fall under the next section.

5 Cf. 0. Birkhoff: Lattice Theory, loc. cit. Chapt. I. In this book orderings, partial
orderings and similar topics are discussed in the spirit of modern mathematics. Exten-
sive references to literature are given there.

590

EXTENSIONS OF THE CONCEPTS

I.e.:

(65:B:a) For any two x y y of D at most one of the three following

relations holds :

x = y, xSy , ySx.

(65:B:b) xSy , y&z together imply xSz.

We call a relation S with these properties a partial ordering of D. 1 Two
x y y of D for which none of the three relations enumerated in (65:B:a) holds
(since the ordering is partial, this is a possibility) are called incomparable
(with respect to S).

Examples of partial orderings are easy to give : The points of the plane,
x&y meaning that the ordinate of x is greater than that of y (cf. footnote 4
on p. 589). We may also define that xSy means that the ordinate and the
abscissa of x are both greater than their counterparts for y . 2 Another good
example obtains in the domain of positive integers, xSy meaning that x is
divisible by y excluding equality.

65 . 3 . 3 . The two preceding concepts of ordering maintained (65:A:b) in
the same form, while (65:A:a) was modified (weakened) to (65:B:a). This
emphasized the importance of (65:A:b), the property of transitivity .* We
will now undertake to weaken the combination of (65:B:a) and (65:A:b)
further, so that (65:A:b) is essentially affected, too.

Note first that (65:B:a) is equivalent to these two conditions:

(65 :C :a) Never xSx.

(65:C:b) Never xSt/, y&x together.

Indeed (65:B:a) excludes these three combinations: x = y y x2>y\
x = y f y&x) xSy, y$x. Now the first and the second are merely two ways
of writing (65:C:a), while the third is precisely (65:C:b).

We now prove:

(65 :D) Consider the assertion:

( A m ) Never xi&ro, £s&ri, • • • , z m Sx m ~ h where x 0 = x m and
Xoy Xiy • • • , x m -i belong to D.

Then we have :

(65 :D :a) (65 :B :a) is equivalent to ( A i) , ( A 2 ) together.

(65:D:b) (65:B:a), (65:A:b) together imply all (Aj), (A 2 ), (A 8 ), • • •

Proof : Ad (65:D:a): Clearly (Ai) is (65:C:a) and (A 2 ) is (65:C:b).
Writing the relations of (A m ) in the reverse order, and applying (65:A:b)
m— I times gives x m Sa:o. As x m = x 0t this means x 0 Sxo f contradicting
(65:B:a).

1 Note that the word partial is used in the neutral sense, i.e., a complete ordering is a
special case of the partial ones, since (65:A:a) implies (65:B:a).

* Note that this is close to a plausible type of partially ordered utilities in the sense of
the last remark of 3.7.2. Each imagined event may be affected with two numerical
characteristics, both of which must be increased in order to produce a clear and repro-
ducible preference.

s Some other important relations, not at all in the nature of an ordering, also possess
this property: E.g. equality, x — y.

THE EXTENSION. SPECIAL CASES

591

This result suggests considering the total aggregate of all conditions
(Ai), (A 2 ), (A 8 ), * * • . They are implied by (65:B:a), (65:A:b), i.e. by
partial ordering, and represent, as will appear, a further weakening of this
property.

We define accordingly:

(65:D:c) A relation S is acyclic if it fulfills all conditions (Ai), (A*),

(A.),

The reader will understand why we call this acyclicity: If any (A m )
should fail, there would be a chain of relations

XiSXo, X 2 SXi, * * * 1,

which is a cycle, since its last element, x m , coincides with its first one, x 0 .

We have already remarked that acyclicity is implied by partial ordering
(this is, of course, the content of (65:D:b)), and hence a fortiori by complete
ordering. It remains to show that it is actually a broader concept than
partial ordering, i.e., that a relation can be acyclical without being an
ordering (partial or complete).

These are examples of the latter phenomenon: Let D be the set of all
positive integers, and xSy the relation of immediate succession, i.e.,
x = y + 1. Or, let D be the set of all real numbers, and xS y the relation ol
being greater than, but not by too much — say by no more than 1 — i.e.,
the relation y + 1 ^ x > y.

We conclude this section by observing that our examples of complete anc
of partial orderings and of acyclical relations could easily be multiplied
Space forbids us to go into this here, but it may be suggested to the readei
as a useful exercise. The references to the literature in footnote 1 on pag<
62 and footnote 5 on page 589 can also be consulted to advantage.

60.4. The Solutions : For a Symmetric Relation. For a Complete Ordering

65 . 4 . 1 . Let us now discuss the schemes of specialization referred to at th(
end of 65.2.

First: S is symmetric in the sense of 30.3.2. In this case it is expedien
to go back to the connection with saturation, pointed out at the beginning o
65.2. Owing to the symmetry of S it will provide all information abou
solutions which we desire.

Second: S is a complete ordering. In this case we define as usual: x is \
maximum of D if no y with ySx exists. It is sometimes convenient to indi
cate the connection with a complete ordering by calling it an absolut
maximum of D . (Cf. this with the corresponding place in the next remark.
Clearly D has either no maximum or precisely one. 1

Now we have:

(65 :E) V is a solution if and only if it is a one-element set, consistin

of the maximum of D.

1 Proof: If x, y are both maxima of D, then y%x and x&y being excluded, (65:A:i
necessitates x — y.

592 EXTENSIONS OF THE CONCEPTS

Proof: Necessity: Let V be a solution. Since D is not empty, V is not
empty either.

Consider a y in V. If xS y, then x cannot be in V, hence a u in V with
uSx exists. The transitivity gives uSy which is impossible, since u , y are
both in V. So no x (in D\) exists with xSt/, 1 and y must be a maximum of D.

So D has a maximum which must be unique (cf. above). Hence V is a
one-element set, consisting of it.

Sufficiency: Let x 0 be the maximum of D, V = (x 0 ). Given a y (of D!),
the validity of x&y for no x of V amounts simply to the negation of x 0 &y.
Since ^Sx 0 is excluded, this negation is equivalent to y = x 0 . So these y
form the set V. Hence V is a solution.

66.4.2. Thus there exists no solution V if D has no maximum, while a
solution exists and is unique if D has a maximum.

If D is finite, then the latter is certainly the case. This is intuitively
quite plausible and also easy to prove. For the sake of completeness and
also to make the parallelism with the corresponding parts of the next remark
more evident, we nevertheless give the proof in full:

(65 :F) If D is finite, then it has a maximum.

Proof : Assume the opposite, i.e., that D has no maximum. Choose
any Xi in D, then an x 2 with x 2 Sxi, then an x 3 with x 3 Sx 2 etc., etc. By
(65:A:b) x m &x n for m > n, hence by (65:A:a) x m ^ x n . I.e., the x h x 2 , x 3 ,

• • • are all distinct from each other, and so D is infinite.

These results show that both the existence and uniqueness of V parallel
those of the maximum of D.

65.5. The Solutions : For a Partial Ordering

66.6.1. Third: S is a partial ordering. In this case we take over literally
the definition of a maximum of D from the preceding remark. It is some-
times convenient to indicate the connection with a partial ordering by
calling it a relative maximum of D. (Cf. this with the corresponding place in
the preceding remark. This contrast is quite useful, footnote 2 below not-
withstanding.) D may have no maximum, it may have one and it may have
several. 2 Thus relative maxima are not necessarily unique, while the
absolute ones are. 3

1 A similar situation was already discussed in 4.6.2.

* The argument of footnote 1 on p. 591 fails, since it depends on (65:A:a) which is now
weakened to (65:B:a).

E.g., take for D the unit square in the plane and define in it a partial ordering by
either one of the two processes in the two first examples at the end of 65.3.2. Then the
maxima of D form its entire upper edge, or the upper and the right edges together,
respectively.

1 The reader is warned against mixing up our notion of a relative maximum with that
one which occurs in the theory of functions: There a local maximum is frequently called
a relative one. Since the quantities involved there are numerical, hence completely
ordered, this has nothing to do with our present considerations.

THE EXTENSION. SPECIAL CASES

593

The question of existence also plays a different role for relative maxima
than for absolute ones. It will appear that the decisive property now is this :

(65 :G) If y in D is not a maximum, then a maximum x with xSy

exists.

For absolute maxima — i.e., if S is a complete ordering — (65:G) expresses
precisely the existence of one. 1 For relative maxima this need not be the
case, i.e. for a partial ordering the mere existence of some (relative) maxima
need not imply (65 :G). Examples of this are easy to give, but we will not
pursue this matter further. Suffice it to say, that (65 :G) will prove to be
the proper extension of the existence of an absolute maximum (cf. the
preceding remark) to the case of relative maxima (cf. below).

Now we have:

(65 :H) V is a solution if and only if (65 :G) is fulfilled (by D and S!)

and V is the set of all (relative) maxima.

Proof: Necessity: Let V be a solution.

If y is not in V, then an a; in V with xSy exists, hence y is not a maximum.
So all maxima belong to V.

If y is in V, then the argument given in the proof of (65 :E) in the preced-
ing remark can be repeated literally, showing that y is a maximum.

So V is precisely the set of all maxima.

If y is not a maximum, i.e., not in V, then an x in V, i.e., a maximum,
with xSy exists, so (65 :G) is fulfilled (by D and S).

Sufficiency: Assume that (65 :G) is fulfilled, and let V be the set of all
maxima.

For x, y in V, xSy is impossible, since y is a maximum. If y is not in V»
i.e., not a maximum, then by (65:G) an x which is a maximum, i.e., in V,
with xSy exists. So V is a solution by (65:1).

The reader should verify how this result (65 :H) specializes to (65 :E) of
the preceding remark when the ordering is complete.

Our result (65 :H) shows that there exists no solution V if D and S do not
fulfill the condition (65 :G), while a solution exists and is unique if this
condition is fulfilled.

65 . 5 . 2 . If D is finite then the latter is certainly the case. We give the
proof in full:

(65:1) If D is finite, then it fulfills the condition (65 :G).

Proof: Assume the opposite, i.e., that D does not fulfill (65 :G). Call a y
exceptional , if it is not a maximum and xSy holds for no maximum x. The
failure of (65 :G) means that exceptional y exist.

Consider an exceptional y. Since it is not a maximum, an x with x$y
exists. Since y is exceptional, this x is not a maximum. If a maximum u

1 Proof: Since D is not empty (65 :G) implies the existence of a maximum.

Conversely: Let Xo be the maximum of D. Then for every y not a maximum, i.e.,
y 9 * Xo , the exclusion of y$x 0 and the validity of (65:A:a) (complete ordering!) give XoSy.

594

EXTENSIONS OF THE CONCEPTS

with u$x existed, this would give by (65:B:b) uSy contradicting the excep-
tional character of y . Hence no such u exists, i.e., x too is exceptional.
I.e.:

(65 :J) If y is exceptional then there exists an exceptional x with x&y.

Now choose an exceptional x h and exceptional £2 with £ 2 §£i, an excep-
tional £3 with £ 8 S >£2 etc., etc. By (65:B:b) £ m S£ n for m > n, hence by
(65:B:a) x m ^ £ n . I.e., £ 1 , £ 2 , £3, * * are all distinct from each other and

so D is infinite.

(Cf . the last part of this argument with the proof of (65 :F) in the preced-
ing remark. Observe, that we could replace its (65:A:a) by the weaker
(65:B:a).)

These results show that the existence of a solution now does not corre-
spond to the existence of a maximum, but to the condition (65 :G). This is
quite remarkable considering the concluding part of the preceding remark
in 65.4.2. It corroborates our earlier observation that in the present case
of partial ordering (65 :G) is the proper substitute for the existence of a
maximum.

The uniqueness of the solution is even more remarkable. In the light
of the last part of our preceding remark, it would have seemed natural for
this uniqueness to be connected with that one of the maximum. But we see
now that the solution is unique, while the (relative) maximum need not be,
as was already mentioned . 1

65.6. Acyclicity and Strict Acyclicity

65 . 6 . 1 . Fourth: S is acyclic. We know that this case comprises the two
preceding ones, i.e., that it is more general than both.

In those two cases we determined the necessary and sufficient conditions
for the existence of a solution and we also found that when they are satisfied
the solution is unique. (Cf. (65 :E) and (65 :H).) Furthermore, it was seen
that when D is finite these conditions are certainly satisfied. (Cf. (65 :F)
and (65:1).)

In the acyclic case we will find conditions which are similar to these in
many ways and in some respects we will gain deeper insights than before.
It will be necessary, however, to vary our standpoint somewhat in the course
of our discussion and our results will be subject to certain limitations. The
case of a finite D will again be settled in an exhaustive and satisfactory way.

It is again convenient to introduce the concept of maxima , 2 and not only
for D itself but also for its subsets. So we define : x is a maximum of E(s D)
if £ belongs to E and if no y in E with y§>x exists. We denote the set of all
maxima of E by E m (z E).

1 ( 65 :H) shows that the solution V is not connected with any particular (non-unique)
maximum, but with the (unique) set of all maxima.

* Since we used the qualification “ absolute” in the second, and “relative” in the third
remark, we should now employ another still weaker one. It seems unnecessary, however,
to bring in such a terminological innovation at this occasion.

THE EXTENSION. SPECIAL CASES 595

Our discussions will show that it is of decisive importance whether D and
§ possess this property:

(65 :K) E ^ Q (for E £ D) implies E m ©.

I.e. : Every non-empty subset of D possesses maxima. 1 Prima facie (65 :K)
does not appear to be related in any way to acyclicity, but there exists
actually a very close connection. Before we attack our proper objective,
the role of solutions in the present case, we investigate this connection.

65.6.2. For this purpose we drop all restrictions concerning D and S, even
that of acyclicity.

It is convenient to introduce a property which is a variation of the ( A m )
of (65 :D) in 65.3.3., and which will turn out intrinsically connected with
them:

(A*) Never XiSx 0) x 2 Sxi, x 8 Sx 2 , • • • , where xo, x h x 2) * * * belong to D. 2
We define, for reasons which shall appear soon:

A relation S is strictly acyclic if it fulfills the condition (AJ.

We now clarify the relationship of strict acyclicity — i.e. of (AJ — both
to (65 :K) and to acyclicity, by proving the five lemmas which follow. The
essential results are (65:0) and (65 :P); (65:L)-(65:N) are preparatory for
(65:0).

(65 :L) Strict acyclicity implies acyclicity.

Proof: Assume that S is not acyclic. Then there exist x 0 , Xi, • ■ • , x m -i
and x m = Xo in Z>, such that XiSx 0 , x 2 Sxi, • • • , x m Sx m -i. Now extend this
sequence x 0 , Xi 9 • • • , x w -i to an infinite one x 0 , Xi, x 2 , * • • by putting

Xo = X m = X 2m == * * * ,

Xl =: Xm-fi = X 2m -f- 1 == * * * t

Xm— 1 — X 2m — i — Xzm— 1 — ■

Then clearly XiSxo, x 2 Sxi, x 8 Sx 2 , • • • etc., etc., and so strict acyclicity fails.

(65 :M) Acyclicity without strict acyclicity implies this:

(J3*) There exists a sequence x 0 , X\ y x 2 , x 8 , • * • * in D with this

property:

1 Even if § is a complete ordering, this property (65 :K) is of great importance in set
theory. Those readers who are familiar with that theory will observe, that (65 :K) is
precisely the fundamental concept of well ordering. (In this case S must be interpreted
as “ before ” instead of “ greater.”) For literature cf. A. Fraenkel , loc. cit. p. 195ff, and
299ff, and F. Hauedorff ’, loc. cit. p. 55ff , both in footnote 1 on p. 61 ; also E. Zermelo loc.
cit. in footnote 2 on p. 269. It is remarkable that the same property plays a role in
connection with our concept of solution for arbitrary relations. The major part of the
considerations which make up the remainder of this chapter deals with this property and
its consequences.

Actually this subject and its ramifications appear to deserve considerable further
study from the mathematical point of view.

* The sequence a?o, x h x if • • • should be infinite in the sense that the indices must go
on ad infinitum , but the x, themselves need not all be different from each other.

1 Cf . this with footnote 2 above, and the last part of this lemma.

596

EXTENSIONS OF THE CONCEPTS

For x p Sx g , p = q + 1 is sufficient and p > q is necessary. 1

( B *) implies that the x 0 , X \ , x 2y * * * are pairwise different
from each other and therefore D must be infinite in this case.

Proof: Since S is not strictly acyclic, there exist x 0 , x ly x 2y * * * in D , such
that XiSx 0 , x 2 Sxi, x 3 Sx 2 , • • • . Hence p = q + 1 is sufficient for x p Sx g .

Now assume that x p Sx q . We wish to prove the necessity of p > q.
Assume the opposite: p ^ q. Now x p +iSx p , x p + 2 &Xp+i y • • • , x fl Sx g _i, 2 *
x p Sx g and these relations contradict ( A m ) with m = q — p + 1: It suffices
to replace its x 0 , Xi f • • • , x m _i and x m = x 0 by our x p , x p +i, • • • , x q
and x p . This conflicts with the acyclicity of S.

Thus all parts of (B*) are established.

Now the consequences of (B*) : If the x 0y X\ y x 2y • • • were not pairwise
distinct, then x v = x q would occur for some p > q. By (B*) x q +i&x qy hence
x q +i&x p ) by (B*) this implies q + 1 > p, i.e. q ^ p y but q < p. So the
Xo y X\ y x 2y * * * are pairwise distinct and therefore D must be infinite.

(65 :N) Non-acyclicity implies this:

For some w(= 1, 2, • • • ) we have:

(B*) There exist x 0 , x iy • • • , x m _i and x m = x 0 in D with this

property:

For XpSx qy p = q + 1 is necessary and sufficient. 8

Proof: Since S is not acyclic, there exist x 0y X\ y • • • , x m -i and x m = x 0
in B such that XiSx 0 , x 2 Sx h • • • , x m Sx m _i. Choose such a system with its
m(= 1, 2, • • * ) as small as possible.

Clearly p = q + 1 is sufficient for x p Sx q . We wish to prove that it is
necessary too. Assume therefore x p Sx q but p q + 1.

Now a cyclical rearrangement of the x 0 , X\ y • • • , x m -\ y x m = x 0 does
not affect their properties and we can apply this so as to make x p the last
element — i.e., to carry p into m. I.e., there is no loss of generality in
assuming p = m. Now p q + 1, i.e., q ^ m — 1. We can also assume
that q j* m y since q = m could be replaced by q = 0. So q ^ m — 2.
After these preparations we can replace Xq , x iy • • • , x m _i, x m = x 0 by
x 0y X\ y • • • , x g , x m = Xo 4 without affecting their properties. This replaces
m by g + 1, which is < m, and this contradicts the assumed minimum
property of m.

Thus all parts of (B*) are established.

65.6.3. Summing up :

(65:0)

(65:0:a) Acyclicity is equivalent to the negation of all (Bf), (Bj),

1 In connection with this result cf. also 65.8.3.

* These are precisely q — p relations, hence they do not appear if p « q.

1 Observe that the characterization of the interrelatedness of the x 0 , X\ y x 2y • • • is
complete in ( B£) y but not in ( B *). This will be of importance below.

4 I.e., omit a?g+i, • • • , *m-i.

THE EXTENSION. SPECIAL CASES 597

(65:0:b) Strict acyclicity is equivalent to the negation of all (Bf),

(£?), • • • and of (£*).

(65:0:c) Strict acyclicity implies acyclicity for all D but it is equiva-

lent to it for the finite D.

Proof: Ad (65:0:a): The condition is necessary since ( B *) contradicts
( Am), hence acyclicity. The condition is sufficient by (65 :N).

Ad (65:0:b): The condition is necessary since non-acyclicity contradicts
strict acyclicity by (65 :L), and (B*) contradicts (4J, hence strict acy-
clity. The condition is sufficient since the negation of strict acyclicity
permits the application of (65 :M) in case of acyclicity, and the application
of (65:0:a) above in case of non-acyclicity.

Ad (65:0:c): The forward implication was stated in (65:L). If D is
finite the reverse implication — and hence the equivalence — results from the
last remark in (65 :M).

Finally we establish the connection with (65 :K):

(65 :P) (65 :K) is equivalent to strict acyclicity.

Proof: Necessity: Assume that S is not strictly acyclic. Choose
Xo,Xi,X 2 , • • * in D with^iSzo, x 2 §X\ y XsSx 2y * * * . Then E = (x 0 , x h x 2y • • * )
is c D and 5 ^ ©, and it possesses clearly no maxima. So (65 :K) fails.

Sufficiency: Assume that (65 :K) fails. Choose a nonempty EsD
without maxima . 1 Choose an x 0 in E. x Q is not a maximum in E , so choose
an Xi in E with XiSx 0 . Xi is not a maximum in E y so choose an x 2 in E with
x 2 Sxi , etc., etc. In this way a sequence Xq> X\ y x 2y • • • in E y hence in D,
obtains and Xi$x 0y x 2 Sx h x 3 Sx 2 , ■ • ■ . This contradicts strict acyclicity.

So we see: Strict acyclicity is the exact equivalent of the property (65 :K),
which we expect to be fundamental. Acyclicity and strict acyclicity are
closely related to each other. The particular role of the finite D begins
already to make itself felt: For finite D the two above concepts are
equivalent.

65.7. The Solutions : For an Acyclic Relation

65 . 7 . 1 . We now turn to our main objective: The investigation of the
solutions in D for S. It is at this point that it will appear, why we attribute
to the property (65 :K) such a fundamental importance: (65 :K) will turn out
to be quite intimately connected with the existence of precisely one solution.

We begin by showing that there exists precisely one solution (in D for $)
if (65 :K) is fulfilled. In proving this we will restrict ourselves to finite
sets D, in which case the solution can even be obtained by an explicit
construction. This construction is effected by finite induction . The finite-
ness of D is not really necessary, but for an infinite set D the construction in
question would be more complicated. 2

1 The reader should compare this proof with that one of (65 :F) in 65.4.2.

2 It would be necessary to make use of more advanced set-theoretical concepts (cf.

598

EXTENSIONS OF THE CONCEPTS

Since we must assume (65 :K), this means by (65 :P) that D must be
strictly acyclic. Since D is finite, this is by (65:0:c) indistinguishable from
ordinary acyclicity. So it does not matter for the moment, whether we state
that we require acyclicity or strict acyclicity of D . It is nevertheless
appropriate to remember that we are using (65 :K), i.e., strict acyclicity,
and that the assumption of finiteness, which obliterates the distinction in
question, could be removed.

We repeat: For the remainder of this paragraph finiteness of D is assumed
and the property (65 :K) — i.e., acyclicity, i.e., strict acyclicity.

Let us now carry out the inductive construction referred to. This will
be done first and the announced properties will be established afterwards.

We define for every i = 1, 2, 3, • • • three sets A», B iy Ci (all sD) as
follows: A i = D. If for an i (= 1, 2, 3, • • • ) Ai is already known, then
Ci and A»+i obtain in this way: Bi = A? i.e., is the set of those y in A {
for which x§> y for no x in Ai. Ci is the set of those y in Ai for which x&y for
some x in Bi. Finally A i+ i = Ai — Bi — Ci.

Now we prove:

(65 :Q) B^ Ci are disjunct.

Proof: Immediate by their definitions.

(65 :R) Ai j* © implies A i+X c A,. 1

Proof: Ai ^ © implies B { = A? j* © by (65:K), 2 hence

Ai+\ = Ai Bi Ci c Ai.

(65 :S) There exists an i with Ai = @.

Proof: Otherwise by (65 :R) D = Ai^ A* 6 A s => • • • , contradicting
the finiteness of D.

(65 :T) Let u be the smallest i of (65 :S), then

D = A x d At => At = • • • => Ai-i ^ Ai o = ©.

Proof: Restatement of (65 :R) and (65 :S).

(65 :U) B i,**, B» # -i, C i, * • • , C< # _ i, are disjunct sets, with

the sum D.

Proof: By the definition of Ai+i we have J5, u C» = A, — A*+i. Hence
Bi u Ci, • • • , Bi - 1 u Ci- 1 , are pairwise disjunct and their sum is

A i — Ai o = D — © — D.

Combining this with (65 :Q) shows that B h C i, • • • , £* # -i, C f# _i, i.e.,

the references of footnote 2 on p. 269 and footnote 1 on p. 595), in particular of transfinite
induction or some equivalent technique.

These matters will be considered elsewhere.

1 The point is that we have c and not merely £!

* This is the only — but decisive! — use we make of (65 :K).

THE EXTENSION. SPECIAL CASES 599

B i, • • • , C i,--*, Ct # -i, are pairwise disjunct, and that their

sum is also D.

65.7.2. We now put

(65:2) Vo « Bi u • • • u B< # - 1 .

Then (65 :U) gives

(65:3) D - Vo = Ci u • • • uC w .

Now we prove:

(65 :V) If V is a solution (in D for S), then V = Vo.

Proof: We begin by showing that £ V for all i = 1 , • • • , i 0 -i.
Assume the opposite and consider the smallest i for which Bi £ V fails
to be true. Let z be an element of this B» not in V. Then ySz for some
y in V. z is a maximum in A» hence y is not in A„ Consider the smallest
k for which y is not in A k . Then k ^ i and as y is in D = Ai, so k t 6 1.
Put j = fc — 1, then 1 ^ j < i. y is in A, but not in A ;+ i = A*, hence it is
in Bj u Cj = A, — A; + i.

z is in Bi&Ai c A,-. So if y were in B„ ySz would imply that z is in C ? .
This is not so, since z is in B x . Hence y is in Cj.

Now necessarily an x in Bj with xSy exists. Since y is in V, this excludes
x from V. Thus Bj £ V cannot hold. As j < i , this contradicts the
assumed minimum property of i.

So we see :

(65:4) BifiV for all i = I, • • * , t 0 — 1.

If y is in C», then an x in B { with x$y exists. Since this x is in V by
(65:4), y cannot be in V.

So we see:

(65:5) Ci s —V for all i = 1, • • • , to — 1.

Comparing (65:4), (65:5) with (65:2), (65:3) above shows that V must
coincide with Vo> as asserted.

(65 :W) Vo is a solution (in D for S).

Proof: We prove this in two steps:

If x, y belong to Vo then xS y is excluded : Assume the opposite: x, y in
Vo, x$y.

x, y belong to Vo, say x to Bi and y to Bj. If i ^ j , then y is
in B, £ A, £ A x . x is in B», so xSy implies that y is in This is not so,
since y is in Bj. If i > j, then x is in £ A» c A,-, y is a maximum in A,-,
hence xSy is impossible.

Thus we have a contradiction in any event.

600

EXTENSIONS OF THE CONCEPTS

If y is not in Vo> then xSy for some x in Vo •' y is in — Vo> hence in some C,.
Hence xSy for an x in Bi , and this x is in consequence in Vo.

This completes the proof.

Combining (65 :V) and (65 :W) we can state:

(65 :X) There exists one and only one solution (in D for S), the Vo of

(65:2) above.

65.8. Uniqueness of the Solutions, Acyclicity and Strict Acyclicity

65 . 8 . 1 . Let us reconsider the last three remarks, still retaining for a
moment the assumption of finiteness, in order to avoid further complications.
It is conspicuous that they all yielded the same result, although under
varying assumptions. In each case we proved the existence of a unique
solution, but the hypothesis was first complete ordering, then partial order-
ing, and finally (ordinary or strict) acyclicity — i.e., it was weakened at
every step.

This being so, it is natural to ask whether we have reached with the last
remark the limit of this weakening — or whether acyclicity could be replaced
by even less without impairing the existence of a unique solution.

It must be admitted, that this line of investigation takes us away from
the theory of games. Indeed, in that theory the existence of solutions was
of primary importance, but we have learned that there could be no question
of uniqueness.

Nevertheless, since we now have some results on existence with unique-
ness, we will continue to study this case. We will see later, that it has even
indirectly a certain bearing on the theory of games. (Cf. 67.)

In the sense outlined we should ask therefore this: Which properties of
the relation S are necessary and sufficient in order that there exist a unique
solution? It is easy to see, however, that this question is not likely to have
a simple and satisfactory answer. Indeed, the solution (in D for S) discloses
only little about the structure of D (together with S). The acyclical case
is less suited to judge this, since it is somewhat complicated, but the cases of
complete or partial ordering make the point quite clear. There the solution
is only related to the maxima of D and it does not express at all what the
properties of the other elements of D are.

It is not difficult to eliminate this objection. Consider a set E&D
instead of D . The relation S in D is also a relation in E and if it was a
complete ordering or a partial ordering or (ordinarily or strictly) acyclic in D,
then it will be the same in E. 1 Hence our result (65 :X) implies that in
every E SiD there exists a unique solution (for S). Now these solutions,
when formed for all E £ D, tell much more about the structure of D . It is
best to restrict ourselves again to the cases of (complete or partial) ordering.
Clearly the knowledge of the maxima of E for all sets E £ D gives a very
detailed information about the structure of D (together with S).

1 I.e., at least the same — it can happen that a partial ordering in D is complete in E —
or that an acyclic relation in D is an ordering in E.

THE EXTENSION. SPECIAL CASES

601

65.8.2. Thus we arrive at the following question: Which properties of
the relation S are necessary and sufficient in order that there exist for each
E zD a unique solution (in E for S)? We can show that here acyclicity
and strict acyclicity are the significant concepts, although the subject is not
completely exhausted. The two lemmas which follow contain what we can
assert on this matter.

(65 :Y) In order that there exist for each E&D a unique solution

(in E for S), strict acyclicity is sufficient.

For finite D this follows from (65 :X), and strict acyclicity
may be replaced by acyclicity, owing to (65:0:c).

For infinite D this is dependent upon the extension of (65 :X)
to infinite sets (cf. the beginning of 65.7.1.)

Proof: If D is (ordinarily or strictly) acyclic, then the same is true of all
E £ D (cf. above). Now all assertions of our lemma become obvious.

(65 :Z) In order that there exist for each E^D a unique solution

(in E for S) acyclicity is necessary.

Proof : If D is not acyclic, then (65:0:a) yields the validity of a ( B *),
m = 1, 2, • • • , in (65:N). Form its x 0 , xi, • • • , x m -i and x m = x 0 and
put E = (x 0 , Xi, x 2 , * * • , x m _i). Then EzD and (B*) describes S in E
completely. Let us consider the solution V in E (for S).

Consider such a solution V. If x» is in V, then x*+i is not, since x,+iSxi.
If x» is not in V then there exists a y in V with t/Sx», i.e., y = x, with x,Sx<.
This means j = i + l, 1 so y = x i+h and hence x t +i is in V. So we see:

(65:6) Xi is in V if and only if x»+i is not.

Iteration of (65:6) gives:

(65 :7) If A; is even, then x 0 is in V if and only if Xk is.

If k is odd, then x 0 is in V if and only if x k is not.

As x 0 = Xm, (65:7) involves a contradiction if m is odd. Hence there
exists no solution in E (for S) if m is odd. If m is even, then (65:7) implies
that V is either the set of all x k with an even k or the set of all x k with an odd
k . And it is easy to verify that both these sets are indeed solutions in
E (for S).

So we have:

(65:8) The number of solutions in E = (x 0 , Xi, • • • , x m -i) for S

(with the x 0 , Xi, • • • , x m _i from (B*)) is 2 or 0 according to
whether m is even or odd.

Consequently there is in no case a unique solution in this E(£ D).
Combining (65 :Y) and (65 :Z) we see: The existence of a unique solution
(in E for S) for all E £ D is completely characterised for finite sets: For these

1 If i — m, then replace it by i * 0.

602

EXTENSIONS OF THE CONCEPTS

it is equivalent to acyclicity, i.e., to strict acyclicity, which in this case is
the same thing. For infinite sets D we can only say that acyclicity is
necessary and strict acyclicity is sufficient.

66.8.3. The gap which exists in this case can only be bridged by a study
of the acyclic, but not strictly acyclic (infinite) sets D and their subsets E .
By comparing (65:0:a), (65:0:b) we see that such a D satisfies (£*).
Form its x 0f x\. x^. • • • and put D* = (x 0 , x h xs, * * • ). This is also
acyclic but not strictly acyclic, hence we may study it in place of D.

Thus the question has become this :

(65:9) Assume that D* = (x 0 , x h x 2 , • • • ) fulfills (£*). Will then

every E &D* possess a unique solution (in E for S)?

The answer to (65:9) cannot be given immediately, because (£*)
describes the relation xSy in D* — i.e. x p Sx q — only incompletely. The cor-
responding question for (£*) (m = 1, 2, • * • ) was answered in the proof
of (65 :Z) in the negative, but (£*) described the relation x$y in its set — i.e.
x p &x q — completely. Thus the answer to (65:9) requires an exhaustive
analysis of all possible forms of the relation x p &x q which fulfill (£*). The
problem appears to be one of considerable difficulty. 1

65.9. Application to Games : Discreteness and Continuity

66.9.1. Our above results on acyclicity and on strict acyclicity have, as
pointed out before, no direct bearing on the theory of games.

As regards strict acyclicity, it suffices to emphasize its equivalence to
(65 :K) (by (65 :P)), and to remember that in the theory of games even D
itself (the set of all imputations) possesses no maxima (i.e., undominated
elements). 2

Ordinary acyclicity too is violated, e.g., already in the essential three
person game. 8

Nevertheless there were situations that arose during the mathematical
discussion of certain games, where the concept of acyclicity could have been
applied. These situations are to be regarded in the spirit of the first remark
of 65.1.1. and they are specifically among the examples referred to there.

Thus in the triangles T discussed in 47.5. 1. we have an acyclical concept of
domination, as the inspection of figures 76, 77 shows. 4 Further in the set Cfc
described in 55.8.2. there is an acyclic concept of domination as the criterion
(55 :Z) makes apparent. 6

Finally, in the market discussed in 64. there is an acyclic concept of
domination in the case of monopoly or monoposony, as the discussion at the

1 It lies on the boundary line of combinatorics and set theory, and seems to deserve
further attention.

* This holds for all essential games. Cf. (31 :M) in 31.2.3.

* The reader is invited to verify this, e.g., on the diagram of Figure 54. It is easy to
ascertain that (£*) holds (and (A m ) fails) for all m gt 3.

4 Here domination implies having a greater ordinate.

1 Here domination implies having a greater n-component, and from this acyclicity
obviously follows.

GENERALIZATION OF THE CONCEPT OF UTILITY 603

end of 64.2.2. and in particular (64:12), (64:13) there shows. 1 We may
accentuate the concluding remark made there by observing that an intrinsic
connection is to be surmised between the monopolistic situations in the eco-
nomic sphere and the mathematical concept of the acyclicity of domination.

It is very remarkable, therefore, that in all these cases particularly exten-
sive families of solutions were found to exist. Indeed, not only numerical
parameters, but even highly undetermined curves or functions entered into
those solutions. For this cf. 47.5.5. and Fig. 81. in the first instance, and the
fifth remark in 55.12. in the second one. In the third instance we can only
refer to the mathematical discussion of a special case: The three-person
market — monopoly versus duopoly — which was analyzed in 62.3., 62.4. and
63.4.

66.9.2. The great number of solutions in the acyclic situations referred
to above may seem natural, if the infiniteness of these D (the set of the
imputations under consideration) is emphasized. After all, it was only for
finite sets D that acyclicity implied uniqueness of the solution, for the
infinite ones strict acyclicity became the crucial concept. (Cf. the last part
of 65.8., in particular 65.8.2.) And all these examples are, of course, not
strictly acyclic, as can be verified with ease.

The situation is nevertheless paradoxical, for the following reason:
Modifications of the concept of utility, which will be considered in 67.1.2.
can be applied in such a manner as to make the sets in question finite.
Then the acyclical games mentioned will have unique solutions. Now these
finite modifications can be made to resemble arbitrarily closely to the origi-
nal, unmodified games. Hence the original acyclic games with many solu-
tions (infinite D\) can be approximated arbitrarily closely by the modified
acyclical games with unique solutions (finite D!). How can the unique
solutions be “ arbitrarily close” approximations of the non-unique ones?

This paradoxical situation will be described in detail in 67. The analysis
which we are going to give there will clarify this lack of continuity and
present an opportunity for some interpretations of a certain interest.

66. Generalization of the Concept of Utility

66.1. The Generalization. The Two Phases of the Theoretical Treatment

66.1.1. In the past sections we have generalized the concept of a solu-
tion — based on a relation S, which takes the role of domination — in a most
extensive way. These generalizations should be used in our theory as
follows: Our concepts of imputation, domination and solutions rest upon the
more fundamental one of utility. Now if we desire to vary the formalism

1 Here the domination implies having a greater 1- (or 1 *-) component, and from this
acyclicity obviously follows.

If neither monopoly nor monopsony exist, i.e., if with the notations loc. cit. l f m > 1,
then (64:10), (64:11) apply instead of (64:12), (64:13) eod. It is easy to verify that in
this case acyclicity does not prevail.

604

EXTENSIONS OF THE CONCEPTS

used to describe the latter, we can try to render these variations adequately
by appropriate generalizations of the former concepts.

Of course, we do not wish to carry out generalizations for their own sake,
but there are certain modifications which would make our theory more
realistic. Specifically: We have treated the concept of utility in a rather
narrow and dogmatic way. We have not only assumed that it is numerical
— for which a tolerably good case can be made (cf. 3.3. and 3.5.) — but also
that it is substitutable and unrestrictedly transferable between the various
players (cf. 2.1.1.). We proceeded in this way for technical reasons: The
numerical utilities were needed for the theory of the zero-sum two-person
game — particularly because of the role that expectation values had to play
in it. The substitutability and transferability were necessary for the theory
of the zero-sum n- person game in order to produce imputations that are
vectors with numerical components and characteristic functions with
numerical values. All these necessities present themselves implicitly in
every subsequent construction built upon the preceding ones — and so in fine
in our theory of the general n-person game.

Thus a modification of our concept of utility — in the nature of a gen-
eralization — appears desirable, but at the same time it is clear that definite
difficulties must be overcome in order to carry out this program.

66 . 1 . 2 . Our theory of games divides clearly into two distinct phases:
The first one comprising the treatment of the zero-sum two-person game and
leading to the definition of its value, the second one dealing with the zero-
sum n-person game, based on the characteristic function, as defined with the
help of the values of the two-person games. We pointed out above, how
each of these two phases makes use of specific properties of the utility
concept. Therefore, if any of these properties are to be generalized, modi-
fied or abandoned, we must study the effect of such a change in each phase.
It is therefore indicated to analyze these two phases separately.

66.2. Discussion of the First Phase

66 . 2 . 1 . The difficulties of generalizing the first phase are very serious.
The theory of the zero-sum two-person game as expounded in Chapter III
makes full use of the numerical character of utility.

Specifically: It is difficult to see how a definite value can be assigned to a
game, unless it is possible for each player to decide in all cases which of the
various situations that may arise is preferable from his point of view. This
means that individual preference must define a complete ordering of the
utilities.

Next the operation of combining utilities with numerical probabilities
cannot be dispensed with either. We have seen that the rules of the game
may explicitly require such operations, if they provide for chance moves.
But even when this is not the case, the theory of Chapter III leads in general
to the use of mixed strategies with the same effect. (Cf. 17.)

GENERALIZATION OF THE CONCEPT OF UTILITY 605

Now it is well known that the completely ordered character of utilities
does not imply the numerical one. But we have seen in 3.5. that complete
ordering in conjunction with the possibility of combining utilities with
numerical probabilities implies the numerical character of utility.

Thus we have at present no way to adscribe a zero-sum two-person game
a value unless numerical utilities are available.

In the n-person game the characteristic function is defined with the
help of the value in various (auxiliary) zero-sum two-person games. Our
reduction of the general n-person games to the zero-sum ones, in addition,
made use of the transferability of utilities from one player to another.

n

Indeed, constructions like 3C n+ i = — £ 3C* in 56.2.2. can hardly be given

fc-i

any other meaning. Thus the definition of the characteristic function in
an n- person game is technically tied up with the numerical nature of utility
in a way from which we cannot at present escape.

The values v(S) of the characteristic function of such a game are the
values of the corresponding sets — coalitions — of players S. Hence our
conclusion can also be stated in this way : Our general method to adscribe a
value to every possible coalition of players is essentially dependent upon
the numerical nature of utility and we are at present not able to remedy this.

We have pointed out before that the hypothesis of the numerical nature
of utility is not as special as it is generally believed to be. (Cf. the discus-
sion of 3.) Besides, we can avoid all conceptual difficulties by referring
our considerations to a strictly monetary economy. Nevertheless it would
be more satisfactory, if we could free our theory of these limitations — and
it must be conceded that the possibility of doing this has not been established
thus far.

66.2.2. In spite of this inadequacy in general, there are many games
where the difficulty of defining the characteristic function is never serious.
Thus the examples of 26.1. and of 57.3. were such that the characteristic
function could be determined directly, without real need for the elaborate
considerations of the theory of the zero-sum two-person games. It is true
that these were examples synthesized in order to obtain a known, pre-
assigned characteristic function — hence the ease with which they can be
handled in this respect is scarcely surprising. However, there exist other
instances of the same phenomenon which are of a certain significance: Thus
the characteristic function causes no difficulty whatever throughout the
theory of simple games of Chapter X. 1 Again the various markets con-
sidered in 61.2.-64.2. all had characteristic functions which were easily and
directly obtained.

In these cases it would be easy to replace the numerical utilities by more
general concepts. We propose to take them up on another occasion.

1 These games were defined by stating which are the winning coalitions and this
implied an implicit determination of the characteristic function.

606

EXTENSIONS OF THE CONCEPTS

66.3. Discussion of the Second Phase

66 . 3 . 1 . If the characteristic function is taken for granted, we can pasi
to the second phase.

Here the necessity for a numerical utility can be entirely circumvented
We do not propose to describe this in complete detail, since the entire
subject does not seem to be mature yet for a final mathematical formaliza
tion. Indeed, the first phase is obstructed by unsolved difficulties ai
described above. Besides, there appears to be some justification for believ
ing that a more unified form of the theory, of which we can at present se<
only the outlines, might lead us to the desired goal.

We shall therefore give only some general indications relative to th<
treatment of the second phase.

To begin with, when we renounce the transferability of utilities, as wel
as when we renounce their numerical character, concepts like zero-sum o]
constant-sum games are not immediately defined. Hence it is best to dea
directly with general games.

Let us therefore consider a general n-person game. Since we possess th<
theory of Chapter XI, we may forget its origin in the theory of zero-sun
games and try to extend it directly to the case of more general (non-numeri
cal, non-transferable) utilities.

The imputations

« = H«i, •••,«.}}

will still be vectors, but their components ai, • • • , a n may not be num
bers. It must be noted, that if we give up the numerical character o
utility, it is best to concede that each participant i( = 1, • • • , n) has t
domain of individual utilities % of his own. I.e., the ^i, • • • , < U» will ii
general be different. In this setup the component a* must belong to %.

It must be noted that even if all utilities are numerical — i.e., if ‘Ui, • • • , %
coincide with each other and with the set of all real numbers — we may stil
omit the assumption of transferability. Also we may consider the cas<
where transferability exists, but subject to certain restrictions. Indeed
an example of this will be discussed in detail in 67.

66 . 3 . 2 . Now the restrictions on these components a» must be considered
They are of two kinds: First the domain of all imputations was defined ii

56.8.2. by

(66:1) ai t v((i)) for i = 1, • • • , n,

n

(66:2) 2 ^ v((l, • • • , »)).»

*-l

Second we defined domination with the help of a concept of effectivit}
based on

1 We prefer to use (56:10) here instead of the alternatively possible (56:25) of (56:I:b
in 56.12.

GENERALIZATION OF THE CONCEPT OF UTILITY 607

(66:3) £ « £ v(S),

tin 8 *

which is (30:3) of 30.1.1.

All these inequalities belong to a common type : A certain set T is given
(T = (t) in (66:1), T = (1, • • • , n) = I in (66:2), T = S in (66:3)) and

the imputation a is required to place the set — coalition — T into a position
which is at least as good (in (66:1)) or at most as good (in (66:2) and in
(66:3)) as that one stated by v(3 n ).

The position of the coalition T — i.e., the composite position of all its
participants — is expressed in all these inequalities by the sum of their com-
ponents: a k . For non-numerical utilities the domains ^i, * • * ^may

kinT

be different from each other, and besides, there may exist no addition in
them — thus rendering formations like ^ a k senseless. But even if the

k in T

utilities are numerical, the use of ^ a k in the above context is clearly

k in T

equivalent to assuming unrestricted transferability. Indeed, the position
of a coalition can be described by the sum of the amounts given to its mem-
bers — without any reference to the individual amounts themselves — only
when those members are able to distribute that sum among themselves in
any way in which they all agree, i.e., if there are no physical obstructions to
transfers.

In general, therefore, we shall have to forego the use of ^ a k . Instead,

k in T

we must introduce the domain of utilities for the composite person, con-
sisting of all members of a given coalition T. Denote this domain by ^(T).
Clearly, C U(( k )) is the same thing as ‘U*. ^( T ) must be obtainable by some

process of synthesis from the ‘V* of all k in T. It is not at all difficult to
devise the proper mathematical procedure required for this process, but we
propose to discuss it on another occasion.

The aggregate of the a k) k in T 7 , as well as the value v(T) of the charac-
teristic function must be elements of this system. The inequalities (66:1),
(66:2), (66:3) refer then to preferences in that system of utilities.

66.4. Desirability of Unifying the Two Phases

66 . 4 . In the hope that the reader will not find the analysis of 66.3. too
sketchy, we now indicate in which way the desired unification of the two
phases may be looked for. Our theory of the zero-sum two-person game
was really based on the same general principles as the subsequent structure
of imputations, domination and solutions for zero-sum n- person games and
even for general n- person games. Specifically, the decisive discussion of
the inter-relatedness of various strategies in a zero-sum two-person game
carried out in 14.5., 17.8., 17.9. — i.e., the analysis of the concept of a good

608 EXTENSIONS OF THE CONCEPTS

strategy — is in many ways analogous to our use of dominations of
imputations. *

Now it would seem that the weakness of our present theory lies in the
necessity to proceed in two stages: To produce a solution of the zero-sum
two-person game first and then, by using this solution, to define a charac-
teristic function in order to be able to produce a solution of the general
n-pergon game, based on the characteristic function. General experience
in mathematics and in the physical sciences indicates that such a two stage
procedure with an intermediary halt — represented in our case by the
characteristic function — has two essential aspects. In the early stages of
the investigation it may be advantageous, since it divides the difficulties.
In the later stages, however, when full conceptual generality is desired, it
can be a handicap. The requirement of producing a sharply defined quan-
tity in the middle of our procedure — in our case the characteristic function —
might be an unnecessary technicality, saddling the main problem with an
extraneous difficulty.

To apply this specifically to our experience with games: We had to
divide the difficulties in order to overcome them and to consider successively
zero-sum two-person games with strict determinateness, zero-sum two-
person games with general strict determinateness, zero-sum n- person games,
general n- person games. However, all these steps but two were finally
merged into the general theory: Only the zero-sum two-person game and
the general n- person game remained. Our insistence on the characteristic
function amounts to insisting that for the zero-sum two-person game an
intermediary result be obtained which is much sharper than that one which
we accepted as satisfactory for the n-person game. 1 Of course, we were
able to fulfill this requirement in the case of a numerical, unrestrictedly
transferable utility. However, this may be different when these assump-
tions concerning utility are discarded. And it seems rather plausible that
our difficulty with the n-person game may be ascribed to our continued
insistence on this special setup for the zero-sum two-person game. Our
present technical procedure forces us to insist in this respect, but this
insistence may nevertheless be misplaced.

A unified treatment for the entire theory of the n- person game — without
the (as it now appears) artificial halt at the zero-sum two-person game and
the characteristic function — may therefore in fine prove to be the remedy
for these difficulties.

67. Discussion of an Example
67.1. Description of the Example

67.1.1. We shall now discuss an example in which the concepts of utility
and transferability are modified. These modifications do not represent a

1 For the zero-sum two-person game we obtained a unique value — i.e., imputation.
For the general n- person game (as well as for the zero-sum one) we had only a — usually
not unique — solution, and even the individual solution is a set of imputations!

DISCUSSION OF AN EXAMPLE

609

particularly significant broadening of our standpoint with respect to those
concepts. The interest of our example is rather that it permits of an
application of our results concerning acyclicity and thereby yields conclu-
sions which throw some new light on the subject discussed at the end of 65.9.
Specifically, it is hoped that procedures of this kind will provide a more
adequate mathematical approach to the phenomenon of bargaining.

67 . 1 . 2 . The modification to be considered is this: We assume that
utility — or its monetary equivalent — is made up of indivisible units. I.e.
we do not question its numerical character but require that its value be — in
appropriate units — an integer. Thus transfers too are necessarily restricted
to integers, but we do not restrict them further. We propose to use the
characteristic function as before, but also with integer values. The con-
cepts of domination and solution, after this, are unaltered.

If this standpoint is applied to general one and two-person games, no
significant changes occur; i.e., everything remains essentially as in our old
theory. It is therefore unnecessary to enter upon the details of these cases.
The three-person game, on the other hand, offers some new features, even
in its old zero-sum form. It gives rise to some quite peculiar difficulties
which appear to be of considerable interest, but are not yet sufficiently
analyzed. We therefore prefer to postpone this discussion for a later
occasion.

This excludes an exhaustive discussion of the general three-person game
in the new setup. We shall, however, analyze a special case which bears
directly upon the nature of bargaining. This is the three-person market,
consisting of one seller and two buyers.

67 . 1 . 3 . We obtained in our previous analysis of this case various solu-
tions, depending on whether we assumed that only one (individual) trans-
action could take place or several, also depending on the relative strength
of the two buyers. These solutions were described in (62 :C) of 62.5.2. and
in (63 :E) of 63.5. In all these cases it appeared that the general solution
was made up of two parts: (62:18) (or (62:20), (62:21), (63:30)) and (62:19)
(or (62:23), (63:31)). Our discussion there showed that the parts of the
type (62:18) correspond to the situation where the two buyers are competing
with each other, while the parts of the type (62:19) correspond to the situa-
tion where they have formed a coalition against the seller. The type (62 :18)
part was uniquely determined and in essential agreement with the ordinary,
common sense economic ideas on the subject. The type (62:19) part, on
the other hand, was defined with the help of some highly arbitrary func-
tional connections. These expressed, as we saw in 62.6.2., the various
possibilities to set up a rule of division between the allied buyers for any
profit obtained. I.e. they constituted their standard of behavior within
their coalition. Our present discussion is going to provide some additional
information concerning the functioning of this part of the social mechanism.

In order to do this effectively, it is reasonable to eliminate from our
problem all those elements which do not contribute to this aspect. I.e. we

610

EXTENSIONS OF THE CONCEPTS

wish to get rid of the type (62:18) part of the solution. We know from
62.5.2., 62.6.1. that this part is of the smallest size — and indeed could be
omitted altogether (cf. footnote 1 on p. 571) — when v = w in the notations
loc. cit. This means that only one (indivisible) transaction can take place
and that the two buyers are of exactly equal strength. The solution is then
given by (62:20) and (62:19) of 62.5., ((62:20) being superfluous, cf. above),
or equivalently by Figure 99.

So we assume v = w in the scheme of 62.1.2. We can simplify the
situation further, without any significant loss, by putting the “ alternative
use for the seller” u = 0. In this way the (62:2)-(62:4) of 62.1.2., defining
the characteristic function, simplify to

( v((l)) = v((2)) = v((3)) = 0,

(67:1) v((l,2)) = v((l,3)) = w, v((2,3)) = 0,

l v((l,2,3)) = w.

The imputations are now defined by

— ►

a = { {«ii aj, <*>}}

with

(67:2:a) ai ^ 0, a* ^ 0, a 8 ^ 0,

(67 :2 :b) + «2 + otz ^ w. 1

67 . 1 . 4 . We now assume all these quantities to be integers — i.e. the
given w and all permissible a h a 2 , of (67:2:a), (67:2:b).

We define domination as before, i.e. following 56.11.1. — which means
that we repeat the definitions of 30.1.1. literally.

It is therefore necessary to determine the character of the sets
Szl = (1,2,3) with respect to their role in defining a domination. It is
easy to show that the sets

S = (1,2), (1,3)

are certainly necessary, and all others certainly unnecessary. 2 Thus we

1 Note that we are using (67:2:b) with ^ and not with =*. This is the standpoint
taken in the discussion of (66:2) in 66.3.2. In the terminology of (56:1 :b) in 56.12., it
amounts to using (56:10) and not (56:25). The reason for this procedure is that the
former condition is the original one (cf., e.g., 56.8.2.), and the equivalence of the two,
made use of in 56.12., fails in the setup to be used now.

It will be seen in the first remark of 67.2.3. that the ^ and the - in (67:2:b) must
produce different results, but that this divergence nevertheless fits into the general
picture. Besides, the use of « instead of ££ in (67:2:b) would lead to results which
differ only in details of secondary importance from those which we are going to obtain.

* The conditions for certainly necessary and certainly unnecessary sets were derived
in 31.1., and reconsidered in 59.3.2. Since our standpoint has changed again (cf. above,
and particularly footnote 1), it would be necessary to reconsider these things once more.
It seems simpler to take them up de novo:

Owing to (67:2:a) above, and the condition (30:3) in 30.1.1., every S with v(<8) — 0
is certainly unnecessary. This disposes of 8 - (1), (2), (3), (2,3). Again (67:1),
(67:2:a), (67:2:b) above give ay + a% £ w ■■ v((l,2)), ay + o* £ w ■■ v((l,3)), hence
8 » (1,2), (1,3) are certainly necessary. And since (31 :C) in 31.1.3. is clearly still valid,
this renders 8 ** (1,2,3) certainly unnecessary.

DISCUSSION OF AN EXAMPLE

611

can use the definition of domination with S - (1,2), (1,3). I.e.:

means that

a H 0

(67:3 :a)

Oil > Pi

and

(67:3:b)

a 2 > P 2 or

Thus domination implies (67:3:a), and therefore it is clearly acyclical.
(Cf. the corresponding discussion of 65.9.) Furthermore, the domain

(67:2:a), (67:2:b) of the a is finite, because the components on, a 2 , a 3 must
be integers. 1

Now we can apply (65 :X) of 65.7.2.: There exists one and only one
solution Vo which is characterized by the formulae (65:2), (65:3), id.

67.2. The Solution and Its Interpretation

67 . 2 . 1 . In order to apply the formulae (65:2), (65:3) of 65.7.2., we
must determine the sets £„ C t defined at the beginning of 65.7.1. Let us
do this for B h C\.

— ► t f — ►

B\ is the set of those a which cannot be dominated. To dominate a

we must increase and a 2 or a 3 without violating (67:2:a), (67:2:b) in

67.1.3. These increases are by 1 at least, while the other one of a 2 , a 8 may

be decreased as far as to 0. Hence a can be dominated, if either

(ai + 1) + (oc 2 + 1) ^ w or (on + 1) + (an + 1) ^ w.

So B i is defined by

(67:4) (ai + 1) + (as + 1) > 10, («i + 1) + (a, + 1) > w.

By (67:2:a), (67:2:b) this implies a 3 < 2, a 2 < 2, i.e. a 2 , a 3 = 0, 1. Now
(67:4) gives, in conjunction with (67:2:a), (67:2:b), the following
possibilities:

(67: A)
(67 :B)
(67 :C)

a 2 = as = 0,

( a 2 = 1, a 3 = 0
or a 2 = 0, a 3 = 1

ai = Wy w — 1;

J> ai = w — 1;

a 2 = a 3 = 1, ot\ = w — 2.

Ci is the set of those a which are dominated by elements of B h — i.e.
by those in (67:A)-(67:C). It is easy to verify that these are characterized

1 This was, of course, not the case in the original continuum setup.

612

by

(67 :D)

EXTENSIONS OF THE CONCEPTS
' «2 = 0 'j

or l, ai ^ w — 2.

a 3 = 0 j

67 . 2 . 2 . Now it is better to deviate from the scheme of (65:2), (65:3) of
65.7.2.; that is, not to continue by determining J5 2 , C 2 , J5s, Ca, • • • , but to
use an inductive process which is better suited to this particular case. This
process goes as follows:

Consider the a with

(67 :E)

<*2 — 0 or as = 0.

They make up exactly (67 :A), (67 :B), (67 :D). We know that among these
Vo contains precisely the (67 :A), (67 :B). The remaining a are those with
(67 :F) a 2 , a 3 ^ 1;

hence undominated by (67 :A), (67 :B). So we form Vo by taking (67 :A),
(67 :B) outside of (67 :F), and repeating the process of finding a solution in
(67 :F).

Compare (67 :F) with (67:2:a), (67:2:b) in 67.1.3. The only difference
is that a 2) « s are increased by 1. Hence w must be treated as if it were
w — 2. Thus Vo now contains further

(67 :G) a 2 = as = 1, oci f= w — 2, w — 3;

f a 2 = 2 , as = 1 \

(67 :H)

or l, a x = w — 3;

<*2 = 1 , ots = 2

and we must repeat the process of finding a solution in
(67:1) a 2 , ^ 2.

Repetition of this procedure assigns

(67 J)
(67 :K)

a 2 = a 3 = 2, ai = w — 4, w — 5;
c*2 = 3, as = 2

or

a 2 = 2, ots = 3

ai = w — 5;

to Vo, and requires us to repeat the process of finding a solution in
(67 :L) a 2 , a 2 ^ 3,

etc., etc.

Thus Vo consists of (67 :A), (67 :B), (67 :G), (67 :H), (67 :J), (67 :K), • • •
This set can be characterized as follows:

(67 :M)

ai = 0, 1, • • • w ;

DISCUSSION OF AN EXAMPLE

613

(67 :N)

w - a i .-

<*2 = a 3 = — o — if w — ai is even;

a 2 = as =

w — 1 — ai
2
or

. _ 10 + 1 - an w — 1 — cki i

(67 :0) { o ' a3 o

if w — ai is odd.

or

a 2 =

w — 1 — ai u> + 1 — an

; a 3 — rt

67.2.3. The results (67:M)-(67:0) suggest these remarks:

First: The values of a x + a 2 + « 8 in this solution are w and w — 1.
Thus we cannot replace the g in (67:2:b) of 67.1.3. by = , the result stated
in (56:I:b) of 56:12 is no longer true. The maximum social benefit is not
necessarily obtained — and this appears as the direct consequence of the
existence o£ an indivisible unit of utility. 1

Second: This “discrete” utility scale converges toward our usual, con-
tinuous one, if w — » qo . (Cf. the corresponding considerations concerning
discrete and continuous “hands” in Poker, in 19.12.) The difference of
ai + a 2 + and w, mentioned above, is at most 1. So it becomes more
and more insignificant as w — > oo , i.e. this aspect of the situation tends to
what it was in the continuous case.

Third: a 2 , «3 differ from each other by at most 1. So this difference too
tends to insignificance as w — > oo . I.e., when we approach the continuous

case the solution tends to look like this:

(67 :P)
(67 :Q)

0 ^ a\ ^ w,

a 2 = a 3

W — a i

2

As pointed out in the first part of 67.1.3., this solution must be compared
with (62:19) in 62.5.1., using the values u = 0, v = w. The two solutions
are indeed similar, but our solution covers only one special case of (62:19):
The monotonic decreasing functions of ai mentioned there coincide with

each other and with — ^ — -

Those functions describe, as discussed in 62.6.2., the rule of division
upon which the two buyers agreed when forming their coalition (which is
expressed by (62:19)). In the continuous case this rule was highly arbi-
trary. But now, in the discrete case, we find that it is completely deter-
mined — the two buyers must be treated exactly alike!

What is the meaning of this symmetry? Are the other distribution
rules — -i.e. the other choices of the functions in (62:19) — really impossible
in the “discrete” case?

1 Cf. this with footnote 3 on p. 513.

614

EXTENSIONS OF THE CONCEPTS

67.3. Generalization : Different Discrete Utility Scales

67.3.1. In order to answer the above questions, we shall try to destroy
the symmetry (between the two buyers), but conserve, the “discreteness.”

This will be done by altering the setup of 67.1. in so far that we assign
the indivisible unit of utility for the buyer 2 a value different from that
one for the buyer 3. Specifically: Let us prescribe that the values of «i, a%
must be integers, while those of a 3 must be even integers. Apart from this,
everything in 67.1. remains unaltered.

We now carry out the equivalent of the considerations of 67.2. Accord-
ingly, we begin by determining the sets B h Ci of 65.7.

Bi is the set of those a which cannot be dominated. To dominate a
we must increase a\ and a 2 or a 3 without violating (67:2:a), (67:2:b) in
67.1.3. These increases are 1 (for a h a 2 ) or 2 (for a 3 ) at least, while the

other one of a 2> a 3 may be decreased as far as to 0. Hence a can be
dominated if either («i + 1) + (a 2 + 1) g w or (cn + 1) + (a 3 + 2) £ w.
So B i is defined by

(67:5) («i + 1) + (a 2 + 1) > («i + 1) + («3 + 2) > w.

By (67:2:a), (67:2:b) this implies a 3 < 2, a 2 < 3, i.e. a 2 = 0, 1, 2, a 3 = 0.
Now (67:5) gives, in conjunction with (67:2:a), (67:2:b), the following
possibilities:

(67 :R) ot 2 = 0, a$ = 0, cti = w, w — 1;

(67 :S) a 2 = 1, as = 0, at = W — 1, w — 2;

(67 :T) a 2 = 2, a 3 = 0, ai = w — 2.

Ci is the set of those a , which are dominated by elements of B x — i.e. by

those in (67:R)-(67:T). It is easy to verify that these are characterized by

(67 :U) a 2 = 0, a x g w — 2;

(67 :V) a 2 = 1, ai £ w - 3.

67.3.2. Now we repeat the variant of 67.2.2.: Instead of determining
B 2) Cj, Bs, C 8 , • • • , we use a different inductive process.

Consider the a with

(67 :W) a 2 = 0, 1.

They make up exactly (67 :R), (67 :S), (67 :U), (67:V). X We know that

among these, Vo contains precisely the (67 :R), (67 :S). The remaining a
are those with

(67 :X) a 2 ^ 2;

1 Note that at cannot be 1, since it must be even.

DISCUSSION OF AN EXAMPLE

615

hence undominated by (67 :R), (67 :S). So we form Vo by taking (67 :R),
(67 :S) outside of (67 :X), and repeating the process of finding a solution in
(67 :X).

Compare (67 :X) with (67:2:a), (67 :2:b) in 67.1.3. The only difference
is that a 2 is increased by 2. Hence w must be treated as if it were w — 2. 1
Thus Vo now contains further

(67 :Y) a 2 = 2, a 8 = 0, ai = w — 2, w — 3;

(67:Z) a 2 = 3, a$ = 0, oti = w — 3, w — 4;

and we must repeat the process of finding a solution in

(67 :A') a 2 ^ 4.

Repetition of this procedure assigns

(67 :B') a 2 = 4, «3 = 0, ai = w — 4, w — 5;

(67 :C') a 2 = 5, a 3 = 0, = w — 5, w — 6;

to Vo and requires us to repeat the process of finding a solution in

(67 :D') a 2 ^ 6,

etc., etc.

Thus Vo consists of (67:R), (67:S), (67:Y), (67:Z), (67:B'), (67:C'),

• • • . This set can be characterized as follows:

(67 :E') = 0, 1, • • • w;

(67 :F') a 2 = w — a\, w — 1 — ai

(excluding the second one when a x = w);
(67 :G') a 3 = 0.

67 . 3 . 3 . The results (67:E')-(67:G') suggest these remarks:

First and second: Concerning the sum ai + + as and its relation to w

we may repeat literally the corresponding parts of 67.2.3.

Third: Here things are altogether different from 67.2.3. We have
identically a 3 = 0. Approaching continuity, i.e. for w — > «> the solution
tends to look like this:

(67 :H') O^ctx^w,

(67 :1') a 2 = u> — ai,

(67 :J') a 3 = 0.

Repeating the comparison to (62:19) in 62.5.1., as made in the cor-
responding part of 67.2.3., we see that the situation is now this: The mono-
tonic functions of (62:19), which describe the rule of division between the
two allied buyers (cf. loc. cit.) are again completely determined — but this
time we find (instead of the equal treatment they received in 67.2.3.) the
entire advantage going to buyer 2!

1 Note the difference between this and the corresponding step in 67.2.2. following
(67 :F) there.

616

EXTENSIONS OF THE CONCEPTS

We must now compare this result with the corresponding one in 67.2.3.
and interpret the entire phenomenon.

67.4. Conclusions Concerning Bargaining

67 . 4 . The conclusion from the results of 67.2.3., 67.3.3. is evident. In
the former case the two buyers had exactly equal powers of discernment —
i.e. equal units of utility — and the rule of distribution was found to treat
them equally. In the latter case buyer 2 had a better power of discernment
than buyer 3 — i.e. 2’s unit of utility was half of 3’s — and in the rule of divi-
sion the advantage went in its entirety to buyer 2. Clearly, if their abilities
had been reversed, the result would have been also. We may also say: The
advantage in the rule of division between allied buyers is equally divided if
they have equally fine utility scales, and goes entirely to the one with the
finer utility scale otherwise. 1

This is true in the discrete case where each participant has a definite
utility scale and the rule of division (i.e. the solution) is uniquely determined.
In the continuous case the “fineness” of the utility scale is undefined and
the rule of division can be chosen in many different ways, as we have seen.

So we observe for the first time how the ability of discernment of a
player — specifically the fineness of his subjective utility scale — has a deter-
mining influence on his position in bargaining with an ally. 2 It is therefore
to be expected that problems of this type can only be settled completely
when the psychological conditions referred to are properly and systemati-
cally taken into account. The considerations of the last paragraph may be
a first indication of the appropriate mathematical approach.

1 It is possible to consider more subtle arrangements: We can assign to a* and to a*
ranges of varying density. In this case we have still a unique solution for the same
reasons as before. The correlation of a 2 , <*« when plotted in the a 2 , as — plane will be a
combination of the three types described above: Symmetric in a 2 , a$ f i.e. parallel to the
bisectrix of the two coordinate axes; parallel to the a r axis; parallel to the a 2 -axis.

It is actually possible to bring about any desired combination of these elements by
choosing the ranges of a 2 and a% appropriately. Any desired shape of the curve can be
approximated arbitrarily well in this manner. In this way the original generality of the
continuous case is recovered.

We do not propose to consider this matter, and various related ones, in detail here.

* This occurs, of course, only when the theory with continuous utilities allows several
different rules of division between allies — which is plainly the case where bargaining plays
a role.

APPENDIX. THE AXIOMATIC TREATMENT OF UTILITY
A.l. Formulation of the Problem

A.1.1. We will prove in this Appendix, that the axioms of utility enumer-
ated in 3.6.1. make utility a number up to a linear transformation. 1 More
precisely: We will prove that those axioms imply the existence of at least
one mapping (actually, of course, of infinitely many) of the utilities on
numbers in the sense of 3.5.1., with the properties (3:1 :a), (3:1 :b); and we
will also prove that any two such mappings are linear transforms of each
other, i.e. connected by a relation (3:6).

Before we undertake this analysis of the axioms (3:A)-(3:C) of 3.6.1.,
two further remarks concerning them may be useful in dispelling possible
misunderstandings.

A.1.2. The first remark is this: These axioms, specifically the group
(3:A), characterize the concept of complete ordering , based on the relations
>, <. We do not axiomatize the relation =, but interpret it as true
identity. The alternative procedure, to axiomatize = also, would be mathe-
matically perfectly sound, but so is our procedure too. The two procedures
are trivially equivalent and represent only variants in taste. The practice in
the relevant mathematical and logical literature is not uniform and we have
therefore adhered to the simpler procedure.

The second remark is this: As pointed out at the beginning of 3.5.1.,
we are using the symbol > both for the “ natural’ * relation u > v affecting
utilities u , v and for the numerical relation p > a affecting numbers p, a;
also we are using the symbol a • • • + (1 — a) * • * both for the “natural”
operation au + (1 — a)v affecting utilities u, v and for the numerical
operation ap + (1 — a)cr affecting numbers p, a (a is a number in either
case). One might object that this practice can lead to misunderstandings
and to confusion; however, it does not, provided that one keeps always in
evidence whether the quantities involved are utilities ( u } v , w) or numbers
(a, 0, y, • • • , p, <t). This identification of the designations for relations
and operations in the two cases (“natural” and numerical) has a certain
simplicity and facilitates keeping track of the “natural” and numerical
pairs of analogs. For these reasons it is fairly generally accepted in similar
situations in the mathematical literature, and we propose to make use of it.

A.1.3. The deductions which follow in A. 2. are rather lengthy and may
be somewhat tiring for the mathematically untrained reader. From the
purely technical-mathematical viewpoint there is the further objection, that
they cannot be considered deep — the ideas that underly the deductions are

1 I.e. without fixing a sero or a unit of utility.

617

618 THE AXIOMATIC TREATMENT OF UTILITY

quite simple, but unfortunately the technical execution had to be somewhat
voluminous in order to be complete. Possibly a shorter exposition might
be found later.

At any rate, we are now forced to use the esthetically not quite satis-
factory mode of exposition which follows in A. 2.

A.2. Derivation from the Axioms

A.2.1. We now proceed to carry out our deductions from the axioms
(3:A)-(3:C) of 3.6.1. The deduction will be broken up into several succes-
sive steps and it will be carried out in this section and the four next ones.
The final result will be stated in (A:V), (A:W).

(A:A) If u < v, then a < fi implies

(1 — a)u + otv < (1 — fi)u + Pv.

Proof: Clearly a = yfi with 0 < 7 < 1. By (3:B:a) (applied to v, v ,
1 — in place of u,v, a) u < (1 — f))u + pv y and hence by (3:B:b) (applied
to (1 — f$)u + 0v y u, 7 in place of u , v , a)

•(1 - p)u + &V > 7 ( ( I - p)u + Pv) + (1 - 7 )u.

By (3 :C :a) this can be written

(1 - p)u + pv > y(Pv + (1 - P)u) + (1 - y)u.

Now by (3:C:b) (applied to v , u } 7, /?, a = y/3 in place of u , v y a, 7 = afi)
the right hand side is av + (1 — a)u y hence by (3:0:a) (1 — a)u + av.
Thus (1 — a)u + av < ( 1 — P)u + fiv, as desired.

(A:B) Given two fixed uo y v 0 with u 0 < Vo , consider the mapping

a —> w = (1 — a)uo + aVo .

This is a one-to-one and monotone mapping of the interval
0 < a < 1 on part of the interval u 0 < w < v 0 . 1

Proof: The mapping is on part of the interval u 0 < w < v 0 : u 0 < w
coincides with (3:B:a) (applied to u 0y v 0} 1 — a in place of u, v, a), w < v 0
coincides with (3:B:b) (applied to v 0y u 0y a in place of u y v y a).

One-to-one character: Follows from the monotony, which we establish
next.

Monotone character: Coincides with (A:A).

(A:C) The mapping of (A:B) actually maps the a of 0 < a < 1

on all the w of u 0 < w < i> 0 .

Proof: Assume that this were not so, i.e. that some w 0 with u 0 < Wo < Vq
were omitted. Then for all a in 0 < a < 1 (1 — a)uo + avo 5^ wo f i.e.

(1 — a)u 0 + av 0 ^ wq. According to whether we have < or >, let a

1 It will appear in (A:C), that this part is actually the whole interval w 0 < w < v<>.

DERIVATION FROM THE AXIOMS 619

belong to class I or II. Thus the classes I, II, which are clearly mutually
exclusive, exhaust together the interval 0 < a < 1. Now we observe:

First: Class I is not empty. This is immediate by (3:B:c) (applied to
uo, Wo, Vo, 1 ~ ot in place of u , w, v , a).

Second: Class II is not empty. This is immediate by (3:B:d) (applied
to Vo, Wo, Uo, a in place of u, w , v , a).

Third: If a is in I and 0 is II, then a < 0. Indeed, since I and II are
disjunct, necessarily a 0. Hence the only alternative would be a > 0.
But then the monotony of the mapping of (A:B) would imply, that since
a is in I, 0 too must be in I — but 0 is in II. Hence only a < 0 is possible.

Considering these three properties of I, II, there must exist an a 0 with
0 < ao < 1 which separates them, i.e. such that all a of I have a g ao,
and all a of II have a ^ ao. 1

Now ao itself must belong to I or to II. We distinguish accordingly:

First: a 0 in I. Then (1 — ot 0 )u 0 + a 0 v 0 < Wo . Alsow 0 < v 0 . Applying
(3:B:c) (with (1 — ao)u 0 + atoVo, Wo, v 0 , y in place of u, w , v , y) we obtain a y
with 0 < y < l and y((l — ao)u 0 + a 0 v 0 ) + (1 — 7)^0 < w 0 , i.e. by (3:C:b)
(with Uo, Vo, y, 1 — a 0 , 1 — a = 7(1 — a 0 ) in place of u , v , a, 0, 7 = a0)
(1 — a)u 0 + av 0 < w 0 . Hence a = 1 — 7(1 — a 0 ) belongs to I. However
a > 1 — (1 — a 0 ) = a 0 , although we should have a ^ a 0 .

Second: a 0 in II. Then (1 — ao)u 0 + a 0 t>o > Wo- Also u 0 < Wo.
Applying (3:B:d) (with (1 — ao)u Q + a 0 v 0 , w 0 , u 0 , 7 in place of u , w , v , a)
we obtain a 7 with 0 < 7 < 1 and 7((1 — a 0 )u 0 + a 0 vo) + (1 — y)u 0 > Wo,
i.e. by (3:C:a) y(a 0 vo + (1 — a 0 )u 0 ) + (1 — 7)^0 > Wo, hence by (3:C:b) (with
Vq, Uo, y, ao, a = 7a 0 in place of u , v , a, 0, y = a0 ) av 0 + (1 — ot)uo > Wo,
i.e. by (3:C:a) (1 — a)u 0 + av 0 > w 0 . Hence a = 7a 0 belongs to II.
However a < a 0 , although we should have a ^ a 0 .

Thus we obtain a contradiction in each case. Therefore the original
assumption is impossible, and the desired property is established.

A.2.2. It is worth while to stop for a moment at this point. (A:B) and
(A:C) have effected a one-to-one mapping of the utility interval uo < w < v 0
( Uo , Vo fixed with u 0 < v 0 , otherwise arbitrary!) on the numerical interval
0 < a < 1. This is clearly the first step towards establishing a numerical
representation of utilities. However, the result is still significantly incom-
plete in several respects. These seem to be the major limitations:

First: The numerical representation was obtained for a utility interval
Uo < w < Vo only, not for all utilities w simultaneously. Nor is it clear,
how the mappings which go with different pairs u 0 , v 0 fit together.

Second: The numerical representation of (A:B), (A:C) has not yet been
correlated with our requirements (3:1 :a), (3:1 :b). Now (3:l:a) is clearly

1 This is intuitively fairly plausible. It is, furthermore, a perfectly rigorous inference.
Indeed, it coincides with one of the classical theorems effecting the introduction of irra-
tional numbers, the theorem concerning the Dedekind cut. Details can be found in
texts on real function theory or on the foundations of analysis. Cf. e.g. C. CaraModory
loc. cit. footnote 1 on p. 343. Cf. there p. 11, Axiom VII. Our class I should be substi-
tuted for the set (a ) mentioned there. The set { A J mentioned there then contains our
class II.

620

THE AXIOMATIC TREATMENT OF UTILITY

satisfied: It is just another way of expressing the monotony that is secured
by (A:B). However the validity of (3:l:b) remains to be established.

We will fulfill all these requirements jointly. The procedure will
primarily follow a course suggested by the first remark, but in the process
the requirements of the second remark and the appropriate uniqueness
results will also be established.

We begin by proving a group of lemmata which is more in the spirit of
the second remark and of the uniqueness inquiry; however it is basic in
order to make progress towards the objectives of the first remark too.

(A:D) Let u 0 , v 0 be as above: u 0 , v Q fixed, u 0 < t> 0 . For all w in

the interval uq < w < v 0 define the numerical function
f(w) = /* 0 , v# (w) as follows:

(i) /(wo) = 0.

(ii) f(v o) = 1.

(iii) f(w) for w ^ u 0 , v 0 , i.e. for u Q < w < v 0} is the number a in
0 < a < 1 which corresponds to w in the sense of (A:B), (A:C).

(A:E) The mapping

w — ► f(w)

has the following properties:

(i') It is monotone.

(ii') For 0 < P < 1 and w ^ u 0

/(( 1 - P)uo + Pw) = Pf(w).

(iii') For 0 < p < 1 and w v 0

/((l - P)v 0 + pw) = 1 - p + Pf(w).

(A:F) A mapping of all w with uo g w ^ v 0 on any set of num-

bers, which possesses the properties (i), (ii) and either (ii') or
(iii'), is identical with the mapping of (A:D)

Proof: (A:D) is a definition; we must prove (A:E) and (A:F).

Ad (A:E): Ad (i'): For uq < w < v 0 the mapping is monotone by (A:B).
All w of this interval are mapped on numbers > 0, < 1, i.e. on numbers >
than the map of uq and < than the map of Vo. Hence we have monotony
throughout u 0 ^ w ^

Ad (ii'): For w = v 0 : The statement is /((l — P)u 0 + Pv 0 ) = /?, and this
coincides with the definition in (A:B) (with p in place of a).

For w t* vo f i.e. u 0 < w < v 0 : Put /( id) = a, i.e. by (A:B)

w = (1 — a)uo + aVo.

Then by (3:C:b) (with r 0 , ^o, P, a in place of u, d, a, p, and using (3:C:a))
(1 — p)uo + Pw = (1 — P)uq + / 3((1 — a)uo + aVo) = (1 — Pa)uo + paVo.
Hence by (A:B) /(( 1 — P)u 0 + Pw) = Pa = Pf(w) y as desired.

Ad (iii'): For w - u 0 : The statement is /(( 1 — P)v 0 + Pu 0 ) = 1 — P,
and this coincides with the definition in (A:B) (with 1 — p in place of a and
using (3:C:a)).

DERIVATION FROM THE AXIOMS

621

For w 9* Uo, i.e. u<> < w < v 0 : Put f(w) = a, i.e. by (A:B)

W — (1 — a)uo + at>o.

Then by (3:C:b) (with u 0 , v 0 , 0, 1 - a in place of u, v, a, 0, and using (3:C:a))
(1 — 0)t>o + 0w = (1 — 0)i>o + 0((1 — a)uo + at>o) = 0(1 — a)u«

+ (1 ~ 0(1 - «))».,

hence by (A:B)

/(( 1 - 0)vo + 0w) = 1 - 0(1 - «) « 1 - 0 + 0 a = 1 - 0 + 0f(w),
as desired.

Ad (A:F): Consider a mapping
(A:l) w— >fi(w)

with (i), (ii) and either (ii') or (iii')- The mapping
(A:2) w^f(w)

is a one-to-one mapping of m<> w ^ t>« on 0 a ^ 1, hence it can be
inverted :

(A:3) a — >^(a).

Now combine (A:l) with (A:3), i.e. with the inverse of (A:2):

(A :4) a —>fi(\fr(a)) = <p(a).

Since both (A:l) and (A:2) fulfill (i), (ii), we obtain for (A:4)

(A:5) ^(0) = 0, *(1) = 1.

If (A:l) fulfills (ii') or (iii'), then, as (A:2) fulfills both (ii') and (iii'), we
obtain for (A :4)

(A:6) <p(0a) = 0<p(a),

or

(A:7) ^(1 - 0 + 0«) - 1 - 0 + 0*>(«).

Now putting a = 1 in (A:6) and using (A:5) gives
(A:8) <p(0) = 0,

and putting a = 0 in (A:7) and using (A:5) gives <p(l — 0) = 1 — 0.
Replacing 0 by 1 — 0 gives again (A:8).

Thus (A:8) is valid at any rate, (ii'), (iii') restrict it to the 0 with
0 < 0 < 1. However (A:5) extends it to 0 = 0, 1 too, i.e. to all 0 with
0 £ 0 £ 1. Considering the definition of <p(a) by (A:3), (A:4), the general
validity of (A:8) expresses the identity of (A:l) and (A:2), which is pre-
cisely what we wanted to prove.

(A:G) Let u 0 , Vo be as above: wo, v 0 fixed, u 0 < v 0 . Let also two

fixed a o, 0o with a 0 < 0o be given. For all w in the interval
uo ^ w iS Vo define the numerical function g(w) — (£|;J|(u>)
as follows:

g(w) = (0o - ao )f(w) -f a 0 ,

(f(w) = /»„,,(«>) according to (A:D)).

622

THE AXIOMATIC TREATMENT OF UTILITY

We note:

(i) g(u 0 ) = a 0 ,

(ii) g(v 0 ) = 0o.

(A:H) This mapping

w -* g(w)

has the following properties:

(i') It is monotone.

(ii') For 0 < 0 < 1 and w ^ u 0

g(( 1 - 0)u o + 0w) = (1 - 0)a o 4- 0 g(w).

(iii') For 0 < 0 < 1 and w 7* Vo

g(( 1 — 0)vo + 0w ) = (1 — 0)0o + 0g(w).

(A:I) A mapping of all w with u 0 £ w £ v 0 on any set of numbers

which possesses the properties (i), (ii) and either (ii') or (iii'),
is identical with the mapping of (A:G).

Proof: Using the correspondence between functions

01 (if) = (00 - «o)/l(tf) + Clo,

or equivalently

/.<») -

P 0 Ot 0

(for f\(w)> gi(w), and also ior f(w), g(w)), the statements of (A:G)-(A:I) go
over into the statements of (A:D)-(A:F). Hence (A:G)-(A:I) follow from
(A:D)-(A:F).

(A:J) Assuming (i), (ii) in (A:G), the equation

g(( 1 - P)u + pv) = (1 - p)g(u) + pg(v)

(u 0 £ u < v ^ vq) with u = Uo, v u 0 is equivalent to (ii')
in (A:I), and with u ^ v 0y v = v 0 it is equivalent to (iii') in
(A:I).

Proof : Ad (ii') : Put u Qy w, p in place of u y v y p.

Ad (iii') : Put w, v 0y 1 — P in place of u, v y p.

A. 2.3. In (A:G)-(A:J) the mapping of a utility interval u 0 ^ w S Vq
on a numerical interval ^ a ^ p 0 has been given its technically adequate
form, with the necessary uniqueness properties. We can now begin to fit
the various mappings

w -> g(w) = gZ'£\(w)

together.

(A:K) Consider and a ifo with u 0 ^ if o ^ fo. Put

yo = 0";:C;(tfo).

Then gZ*% |(tt>) coincides with 0Cj;2)(if) in the latter’s domain
Wo Si if if o (if ifo 5^ uo, i.e. u<> < wo), and 0*j£j(tf) coincides

DERIVATION FROM THE AXIOMS

623

with gl * in the latter’s domain w 0 ^ w v 0 (if w 0 ^ v 0>
i.e. Wo < v®).

Proof: Ad gl\'Z\(w): gl£\(w) possesses the properties (i), (ii') (of (A:G),
(A:H)) for a 0 , 7o, u 0 , w 0 , because they coincide with those for a 0 , 0o, u 0 , v 0
(since they involve only the lower end a 0 , u 0 ). It also possesses (ii) (of
(A:G)) for a 0 , 70, u®, wo, because fJ!j;Jj(wo) * 70. Hence it follows from
(A:I) that gZ *{ J fulfills within u% £ w £ Wo a unique characterization of

Ad gZ % *\'- gZ*i\ possesses the properties (ii), (iii') (of (A:G), (A:H))for
70, 0o, wo, v 0 because they coincide with those for a 0 , 0o, u 0 , Vo (since they
involve only the upper end 0®, t> 0 ). It also possesses (i) (of (A:G)) for
70, 0o, Wo, v 0> because gZfi 9 (wo) * 70. Hence it follows from (A:I) that gZ\Z\
fulfills within Wo & w £ t>® a unique characterization of gZ\\ J*

(A:L) Consider a gZrf,\ and two u x , v x with u 0 ^ tti < v x ^ v 0 .

Put on « gZ\:i\(ui)i 0i * gZ'*\(vi). Then gl'*\(w) coincides
with gZ j;f|(w) in the latter’s domain ui ^ w ^ t>i.

Proof: Apply first (A:K) to and gZ*Z J (i.e. with it 0 , t>o, a 0 , 0o, v Xy 0i in

place of u 0 , v 0 , a 0 , 0o, w 0 , note that 0i = gZl^(vi)) — this shows that
gZ$l(w) coincides with fif“^(io) in the latter’s domain u 0 ^ w ^ v x . Apply
next (A:K) to gZ\*\ and gZ\f*\ (i.e. with u 0 , v x , a 0 , 0i, Mi, <*1 in place of Uo, v 0 ,
oto, 00, Wo, 7o; note that «1 = flC;:!;(ui) = ^;(u0)— this shows that ^;(io),
and hence also flCj;Jj(w), coincides with gZ |£|(w) in the latter’s domain
u x ^ w Vi.

(A:L) has to be combined with a second line of reasoning. At this
point we also assume that two it*, v* with u* < v* have been chosen; we
will consider them as fixed from now on until we get to (A:V) and (A:W).

We now prove :

(A:M) If it 0 £ u* < t>* ^ Vo, then there exists one and only one

0C|’,Jj(w) such that

(1) = o,

(ii) rf&O V*) * 1.

We denote this flC;:J|(tc) by h UvU (w).

Proof : Form the/(u>) = /„,„(«>) of (A:D). As u* < t>*, so/(m*) < /(»*).
For variable ao, /3o (A:G) gives gl*H\(w) = (0e — <*o )/(u>) + «o- Hence the
above (i), (ii) mean that (0 O — a 0 )/(«*) + «o = 0, (0o — ao)/(t>*) + “o = 1,
and these two equations determine a#, 3o uniquely. 1 Hence the desired

"* “ /(»*) -/(V) * /(•*) -/(«*)

624

THE AXIOMATIC TREATMENT OF UTILITY

g exists and is unique.

(A:N) If uo ^ U\ ^ u* < v* g v\ g wi, then h UotVfi (w) coincides

with h UvVi (w) in the latter’s domain u\ g w g Di.

Proof : Put ax = K vU (ui), /Si = K o%Vq (v x). Then, by (A:L), h » 0 , Vo (w)
coincides with gZ\\t]( w ) in the latter’s domain u\ ^ w g Di. Applying this
to id = u* and to id = v* gives gZ\',v\(u*) = h Uo , V(i (u*) = 0 and < 7 “|;*|(d*) =
K t ,vS v *) ~ 1- Hence by (A:M), (/^'(id) = h UvVi (w). Consequently
h UotVll (w) coincides with h UvVi (w) in the latter’s domain Ui ^ w ^ Di.

We can now establish the decisive fact: The functions h U9tV<t (w) all fit
together to one function. Specifically :

(A:0) Given any w , it is possible to choose Uo> d 0 so that Uo ^ u*

<v Do and id ^ Do- For all such choices of u 0 > d 0 ,

h u „vS w ) ^ as the same value. I.e. h u ^ Vo (w) depends on w only.

We denote it therefore by h(w).

Proof : Existence of u 0 , d 0 : u 0 = Min (u*, w) and d 0 = Max (d*, w)
obviously possess the desired properties.

h UrV9 (w) depends on w only: Choose two such pairs iz 0 , d 0 and u’ 0} dJ:
Uo ^ u* < v* ^ D 0 , Uo ^ w ^ Vo and u r 0 ^ u* < v* g dJ, ?/q ^ w ^ v f .

Puti/x = Max (u 0 , wj), Di = Min(D 0 , dJ). Thenw 0 ^ U\ ^ u* < v* ^ v Y ^ d 0 ,

U\ S w ^ Di, and ^ w, g n* < £ Di ^ d£, Wi ^ id ^ Dj. Now two

applications of (A:N) (first with u 0} d 0 , u h v u w } then with u' 0 , v' 0 , u h v u w)
give K^ 9 (w) = K vVy (w) and h u ; tV ;(w) = h Ujt9i (w). Hence

K„v.( w> ) =

as desired.

A.2.4. The function A(id) of (A:0) is defined for all utilities and it has
numerical values. We can now show with little trouble that it possesses all
the properties that we need.

This is most easily done with the help of two auxiliary lemmata.

(A:P) Given any two v with u < d, there exist two uo , d 0 with

Uo ^ U* <D* ^ D 0 , Uo ^ U < V g D 0 .

Proof: Put tz 0 = Min (i**, m), d 0 = Max (d*, d).

(A:Q) Given any two u , v with u < v, put h(u) = a, /i(d) = 0.

Then a < f), and /i(id) coincides with gZ\v(w) in the latter’s
domain u £ w £ v.

Proof: Choose u 0 , v 0 as indicated in (A:P). By (A:M) K 9tVo (w) is a
9ufy( w ) with two suitable a 0 , /3 0 . By (A:0) h(w) coincides with h Utti , 9 (w),
i.e. with gVfv\( w )> in the latter’s domain uo £ w Do. Applying this to

DERIVATION FROM THE AXIOMS

625

w = u and to w = v gives gZ r f v \{u) = h(u) = a and gZ**\(v) = h{v) * 0.
Since 0*j;Jj(w) is monotone, this implies a < 0. Next by (A:L) (with
w 0 , Vo, «o, 0o, w, v, a, P in place of w 0 , v 0 , «o, Po, u h v h a h Pi)gu^ 9 M coincides

with qZ'H(w) in the latter's domain u ^ w ^ v. Consequently the same is
true for h(w).

After these preparations we establish the relevant properties of h(w)
(A:R) The mapping

w — > h(w)

of all w> on a set of numbers has the following properties:

(i) h(u*) = 0.

(ii) h(v*) * 1.

(iii) /i(w>) is monotone,

(iv) For 0 < y < 1 and u < v

h({ 1 - y)w + yv) = (1 - 7)M W ) +

(A:S) A mapping of all w on any set of numbers, which possesses

the properties (i), (ii) and (iv) is identical with the mapping of
(A:R).

Proof : Ad (A:R): Ad (i), (ii): Immediate by (A:0) and (A:M).

Ad (iii): Contained in (A:Q).

Ad (iv) : Choose w, v according to (A:P) and then a, P and gu,v( w ) accord-
ing to (A:Q). Now by (A:H), (ii') (with w, v, v } y in place of w 0 , v 0 , w, y)
- y)u + yv) = (1 - y)gZ'Z(u) + y gZi(v). Hence by (A:Q) •

M(1 ~ y)u + yv) = (1 — y)h(u) + yh(v)

as desired.

Ad (A:S): Consider a mapping

w — ► hi(w)

of all utilities w on numbers, which fulfills (i), (ii) and (iv). Choose two
w 0 , v 0 with wo g u* < v* ^ v 0 , and put a 0 = hi(u*), Po = hi(v*). Then,
by (A:I), hi(w) coincides with gZ 9 9 ,v° 9 ( w ) in the latter's domain w 0 g w g v 0 .
Putting w = u* and w = v* we get 0«j;(Jj(w*) = hi(u*) = 0, 0Sj£j(v*) =
h\{v*) = 1. Hence by (A:M) gZ 9 ft* 9 is Thus h\(w) coincides with

h u% , u (w) f i.e. with h(w), in w 0 ^ w ^ v 0 . By (A:0) this means that hi(w)
and h(w) are altogether identical.

A.2.5. (A:R), (A:S) give a mapping of all utilities on numbers, which
possesses plausible properties and is uniquely characterized by them, and
therefore we might let the matter rest there. However, we are not yet

626

THE AXIOMATIC TREATMENT OF UTILITY

quite satisfied, for the following reasons: The characterization in (A:R) does
not coincide with that one by (3:l:a), (3:l:b) — (A:R) goes less far in (iv)
(this is asserted in (3:1 :b) for all u, v, in (iv) only for those with u < v);
and it introduces an arbitrary normalization in (i), (ii) (by means of the
arbitrary u*, v*). In what follows, we will eliminate these maladjustments.
This will prove fairly easy.

We first extend (iv) in (A:R).

(A:T) Always (1 — y)u + yu = u.

Proof: For u £ (1 — y)u + yu say that y belongs to class I (upper case)
or II (lower case). If y is in class I or II and if 0 < p < 1 , then

u $ (1 - p)u + p((l - y)u + yu) £ (1 - y)u + yu

by ( 3 :B:a) and ( 3 :B:b). (For y in class I or II, respectively: First, u,
(I — y)u + yu, 1 — /S in place of u, v , a in (3:B:a) or (3:B:b). Second,

(1 — y)u + yu, u , P in place of u, v, a in (3:B:b) or (3:B:a).) By(3:C:a)

and (3:C:b) (with u, u, P, y in place of u, v , a, P)

(1 - f?)u + 0((1 - y)u + yu) * (1 - fiy)u + Pyu.

Hence u £ ( 1 — fiy )u + Pyu ^ (1 — p)u + Pu. Put 8 = Py. Since p is
free in 0 < p < 1, therefore 8 is free in 0 < 8 < y. Assuming 0 < y < 1,
0 < 8 < 1 , we have therefore:

(A:9) If 7 is in class I or II, then every 8 < 7 is in the same class I

or II.

(A:10) Under the conditions of (A:9)

(1 — 8)u + 8u £ (1 — 7 )u + yu,
respectively.

The expression (1 — 7 )u + yu is unchanged if we replace 7 by 1 — 7 .
Asl — 7 <l — $is equivalent to 7 > 8, we can put 1 — 7 , 1 — 8 in place
of 7 , 8 in (A:9). Then (A:9) and (A:10) become this:

(A:ll) If 7 is in class I or II, then every 8 > 7 is the same class I

or. II.

(A:12) Under the condition of (A:ll)

(1 — 8)u + 8u ^ (1 — 7 )u + yu,
respectively.

Now (A:9) and (A:ll) show, that if 7 is class I or II, then every $(< 7
or = 7 or >8) is in the same class I or II. I.e. if either class I or II is not
empty, then it contains all 8 with 0 < 8 < 1 . Assume this to be the case
(for class I or II), and consider two 7 , 8 with 7 <8. Then by (A:10)
(1 — 8)u + 6u $ (l — 7 )u + yu, and by (A: 12 ) (with 8, 7 in place of 7 , i)

DERIVATION FROM THE AXIOMS

627

(1 — 6)u + lu ^ (1 — y)u + 7 u. Hence at any rate both < and > hold
in (1 — 6)u + 6u ^ (1 — 7 )u + 7 u. This is a contradiction. Therefore
both classes I and II must be empty.

Consequently never t* £ (1 — y)u + yu , i.e. always (1 — y)u + y u « w,
as desired.

(A:U) Always

A((1 *“ y)u + yv) = (1 ~ y)h(u) + yh(v)

(0 < y < 1, any u, v).

Proof: For u < v this is (A:R), (iv). For u > v it obtains from the
former by putting v y u> 1 - y in place of u, v, y. For u = v it follows from
(A:T).

We can now prove the existence and uniqueness theorem in the desired
form, i.e. corresponding to (3:1 :a) and (3:l:b). At this point we also drop
the assumed fixed choice of u*, v *, which was introduced before (A:M).

(A:V) There exists a mapping

w —> v(w)

of all w on a set of numbers possessing the following properties:

(i) Monotony.

(ii) For 0 < 7 < 1 and any u , v

v((l - 7 )u + yv) = (1 - 7 )v(u) + yv(v).

(A:W) For any two mappings v(w) and v'(w) possessing the prop-

ties (i), (ii), we have

v'(ttf) = 0>oV(tt>) + «1,

with two suitable but fixed u> 0 , wi and w 0 > 0.

Proof: Let u*, v* be two different utilities, 1 u* $ v*.

If u * > v*, then interchange u* and v*. Thus at any rate u * < v*.
Use these u *, v* for the construction of h(w) y i.e. for (A:L)-(A:U). We now
prove :

Ad (A:V)-: The mapping

w — ► h(w)

fulfills (i) by (A:R), (iii), and (ii) by (A:U).

Ad (A:W); Consider v(w) first. By (i) v(u*) < v(t>*)- Put

hi(w)

v{w) — v(u*)
v(t>*) — v(u*)

1 Strictly speaking, the axioms permit that there should be no two different utilities.
This possibility is hardly interesting, but it is easily disposed of. If there are no two
different utilities, then their number is sero or one. In the first case our assertions are
vacuously fulfilled. Assume therefore the second case: There exists one and only one
utility t0 o . A function is just a constant v(u>o) *■ «o- Any such function fulfills (i), (ii)
in (A:V). In (A:W), with v(u>) - ac, v'(w) — choose o> 0 ■* 1 and «i - aj —

628

THE AXIOMATIC TREATMENT OF UTILITY

Then hi(w) fulfills (i), (ii) in (A:R) automatically, and (iii), (iv) in (A:R) by
(i), (ii) above. Hence by (A:S) hi(w) = h(w), i.e.

(A:13) v(w) = ao h(w) + c*i,

wheTe a 0) cti are fixed numbers: a 0 = v(0 — v(w*) >0, on = v(u*).
Similarly for v'(w ) :

(A:14) \'(w) = a f 0 h(w) + «!,

where are fixed numbers: a' 0 = v'(v*) — v'(w*) > 0 , a[ = v(i/*).

Now (A:13) and (A:14) give together

(A:15)

v'(w) = G)ov(w) + 0)1,

where ojo, o)i are fixed numbers: wo = — > 0, m = Hi-** This is

«0 ao

the desired result.

A.3. Concluding Remarks

A.3.1. (A:V) and (A:W) are clearly the existence and uniqueness
theorems called for in 3.5.1. Consequently the assertions of 3. 5.-3. 6. are
established in their entirety.

At this point the reader is advised to reread the analysis of the concept
of utility and of its numerical interpretation, as given in 3.3. and 3.8. There
are two points, both of which have been considered or at least referred to
loc. cit., but which seem worth reemphasizing now.

A.3.2. The first one deals with the relationship between our procedure
and the concept of complementarity. Simply additive formulae, like
(3:1 :b), would seem to indicate that we are assuming absence of any form
of complementarity between the things the utilities of which we are combin-
ing. It is important to realize, that we are doing this solely in a situation
where there can indeed be no complementarity. As pointed out in the first
part of 3.3.2., our u, v are the utilities not of definite — and possibly coexist-
ent — goods or services, but of imagined events. The u, v of (3:l:b) in
particular refer to alternatively conceived events u> v , of which only one can
and will become real. I.e. (3:1 :b) deals with either having u (with the
probability a) or v (with the remaining probability 1 — a ) — but since the
two are in no case conceived as taking place together, they can never com-
plement each other in the ordinary sense.

It should be noted that the theory of games does offer an adequate way
of dealing with complementarity when this concept is legitimately appli-
cable: In calculating the value v(S) of a coalition S (in an n-person game),
as described in 25., all possible forms of complementarity between goods or
between services, which may intervene, must be taken into account. Fur-
thermore, the formula (25:3:c) expresses that the coalition S u T may be
worth more than the sum of the values of its two constituent coalitions S
T, and hence it expresses the possible complementarity between the services

CONCLUDING REMARKS

629

of the members of the coalition S and those of the members of the coalition
T. (Cf. also 27.4.3.)

A.3.3. The second remark deals with the question, whether our approach
forces one to value a loss exactly as much as a (monetarily) equal gain,
whether it permits to attach a utility or a disutility to gambling (even when
the expectation values balance), etc. We have already touched upon these
questions in the last part of 3.7.1. (cf. also the footnotes 2 and 3 eod.).
However, some additional and more specific remarks may be useful.

Consider the following example: Daniel Bernoulli proposed (cf. footnote
2 on p. 28), that the utility of a monetary gain dx should not only be pro-
portional to the gain dx, but also (assuming the gain to be infinitesimal —
that is, asymptotically for very small gains dx) inversely proportional to the
amount x of the owner's total possessions, expressed in money. Hence

dx

(using a suitable unit of numerical utility), the utility of this gain is —

Xl f dx x

The excess utility of owning x h over owning x 2 , is then / — = In ~

xtj % X 2

The excess utility of gaining the (finite) amount rj over losing the same

amount is In In — ^ — = In ( 1 — \ )• This is < 0, i.e. of equal

X XT) \ x f

gains and losses the latter are more strongly felt than the former. A
50%-50% gamble with equal risks, is definitely disadvantageous.

Nevertheless Bernoulli's utility satisfies our axioms and obeys our results:
However, the utility of possessing x units of money is proportional to In x,
and not to xl 12

Thus a suitable definition of utility (which in such a situation is essen-
tially uniquely determined by our axioms) eliminates in this case the specific
utility or disutility of gambling, which prima facie appeared to exist.

We have stressed Bernoulli's utility, not because we think that it is par-
ticularly significant, or much nearer to reality than many other more or less
similar constructions. The purpose was solely to demonstrate, that the use
of numerical utilities does not necessarily involve assuming that 50%-50%
gambles with equal monetary risks must be treated as indifferent, and the
like.*

It constitutes a much deeper problem to formulate a system, in which
gambling has under all conditions a definite utility or disutility, where
numerical utilities fulfilling the calculus of mathematical expectations cannot
be defined by any process, direct or indirect. In such a system some of our

1 The 50%-50% gamble discussed above involved equal risks in terms of x, but not in
terms of In x.

1 That the utility of x units of money may be measurable, but not proportional to x }
was pointed out in footnote 3 on p. 18.

1 As stated in remark (1) in 3.7.3., we are disregarding transfers of utilities between
several persons. The stricter standpoint used elsewhere in this book, as outlined in
2 . 1 . 1 ., specifically, the free transferability of utilities between persons, does force one to
assume proportionality between utility and monetary measures. However, this is not
relevant at the present stage of the discussion.

THE AXIOMATIC TREATMENT OF UTILITY

axioms must be necessarily invalid. It is difficult to foresee at this time,
which axiom or group of axioms is most likely to undergo such a modification.

A.3.4* There are nevertheless some observations which suggest them-
selves in this respect.

First: The axiom (3:A) — or, more specifically, (3:A:a) — expresses the
completeness of the ordering of all utilities, i.e. the completeness of the
individual's system of preferences. It is very dubious, whether the idealiza-
tion of reality which treats this postulate as a valid one, is appropriate or
even convenient. I.e. one might want to allow for two utilities u , v the
relationship of incomparability y denoted by u || v , which means that neither
u = v nor u > v nor u < v. It should be noted that the current method of
indifference curves does not properly correspond to this possibility. Indeed,
in that case the conjunction of “ neither u > v nor u < v ,” corresponding
to the disjunction of “ either u = v or u || v ” and to be denoted by u « v y can
be treated as a mere broadening of the concept of equality (of utilities, cf.
also the remark concerning identity in A. 1.2.).

Thus if u || u' t v || r', then u', v' can replace u, v in any relationship, e.g.
in this case u < v implies u ' < v'. Hence in particular u || u ' and v = v'
have this consequence, and u = u' and v || v' have this consequence. I.e.,
writing v , w, n for w, v , v! and u , v , w for u , v, v\ respectively:

(A:16) u || v and v < w imply u < w.

(A:17) u < v and v || w imply u < w.

However, for the really interesting cases of partially ordered systems
neither (A:16) nor (A:17) is true. (Cf. e.g. the second example at the end
of 65.3.2., which is also dealt with in footnote 2 on page 590, where the
connection with the concept of utility is pointed out. This is the ordering
of a plane so that u > v means that u has a greater ordinate than v as well
as a greater abscissa than v.)

Second: In the group (3:B) the axioms (3:B:a) and (3:B:b) express a
property of monotony which it would be hard to abandon. The axioms
(3:B:c) and (3:B:d), on the other hand express what is known in geometrical
axiomatics as the Archimedean property : No matter how much the utility
v exceeds (or is exceeded by) the utility u, and no matter how little the
utility w exceeds (or is exceeded by) the utility u, if v is admixed to u
with a sufficiently small numerical probability, the difference that this
admixture makes from u will be less than the difference of w from u. It is
probably desirable to require this property under all conditions, since its
abandonment would be tantamount to introducing infinite utility differences. 1

1 For a statement of the Archimedean property in an axiom atisation of geometry,
where it originated, cf. e.g. D. Hilbert , loc. cit. footnote 1 on page 74. Cf. there Axiom
V.l. The Archimedean property has since been widely used in axiomatizations of
number systems and of algebras.

There is a slight difference between our treatment of the Archimedean property and
its treatment in most of the literature we are referring to. We are making free use of the
concept of the real number, while this is usually avoided in the literature in question.
Therefore the conventional approach is to “majorise” the “larger” quantity by successive

CONCLUDING REMARKS

681

In this connection it is also worth while to make the following observa-
tion: Let any completely ordered system of utilities be given, which does
not allow the combination of events with probabilities, and where the
utilities are not numerically interpreted. (E.g. a system based on the
familiar ordering by indifference curves. Completeness of this ordering
obtains, as indicated in the first remark above, by extending the concept of
equality — i.e. by treating the concept u « v , that we introduced there, as
equality. In this case u ® v means, of course, that u and v lie on the same
indifference curve.) Now introduce events affected with probabilities.
This means that one introduces combinations of, say, n ( = 1,2, • • •) events

»

with respective probabilities a h • • • , a n (an, • • • , a n ^ 0, £ a* = 1).

*-i

This requires the introduction of the corresponding (symbolic) utility
combinations a\U\ + • • • + ot n u n (u h • , u n in *u). It is possible to
order these a\U X + • * + ot*u n (any n = 1,2, • • • and any <*i, • • • , a n

and ui f • • • , u n subject to the above conditions) completely, and without
making them numerical — if the ordering is allowed to be non-Archimedean.
Indeed, comparing, say, cnui + • • • + a n u n and p x v x + • • • + p m v m
we may assume that n = m and that the u h • • • , u n and the v\, • • • , v m

coincide (write cnui + • • • + oc n u n + 0 v x + • • • + 0 v m and Oui + • • •

+ 0 Un + /SiVi + • * + fl m Vm fOT a\U\ + * * + Ot n U n and piVi + * * * +

ff m v mt and then replace n + m; u h • • • , u n , v h • • • , v m ; a h • • • , a n , 0,

} ' ' ' ) l^l> > by fl , til, > ti n J Otl, > OCn } j8l,

* * * , fin). Then we compare «iUi + * • • + a n u n and p x Ui + • • • +
PnUn. Next make, by an appropriate permutation of 1, • • • , n, tii >

* * * > Un. After these preparations define aiUi + • • • + a n u n > PiUi +

* • • + 0 n u n as meaning that for the smallest i(=l, • • • , n) for which

a i 9 * say i * t 0 , there is a i§ > p i$ .

It is clear that these utilities are non-numerical. Their non-Archi-
medean character becomes clear if one visualizes that here an arbitrary
small excess probability affecting u <# will outweigh any potential

opposite excess probabilities Pi — a, of the remaining u t , t = to + 1 , * * * , u,
i.e. of utilities < u l# . (This then excludes the application of criteria like
that one in footnote 1 on page 18.) Obviously, they violate our axioms
(3 :B :c) and (3:B:d).

Such a non-Archimedean ordering is clearly in conflict with our normal
ideas concerning the nature of utility and of preference. If, on the other

additions of the "smaller” one (cf . e.g. Hilbert's procedure loc. cit.), while we "minorise”
the "smaller” entity (the utility discrepancy between w and u in our case) by a suitable
small numerical multiple (the a-fold in our case) of the "larger” entity (the utility
discrepancy between v and u in our case).

This difference in treatment is purely technical and does not affect the conceptual
situation. The reader will also note that we are talking of entities like "the excess of
v over u” or the "excess of u over v” or (to combine the two former) the "discrepancy of
u and t>” (u, v, being utilities) merely to facilitate the verbal discussion — they are not
part of our rigorous, axiomatic system.

632 THE AXIOMATIC TREATMENT OF UTILITY

hand, one desires to define utilities (and their ordering) for the probability-
including system, satisfying our axioms (3:A)-(3:C) — and hence possessing
the Archimedean property — then the utilities would have to be numerical,
since our deduction of A. 2. applies.

Third: It seems probable, that the really critical group of axioms is
(3:C) — or, more specifically, the axiom (3:C:b). This axiom expresses
the combination rule for multiple chance alternatives, and it is plausible,
that a specific utility or disutility of gambling can only exist if this simple
combination rule is abandoned.

Some change of the system (3:A)-(3:C), at any rate involving the
abandonment or at least a radical modification of (3:C:b), may perhaps lead
to a mathematically complete and satisfactory calculus of utilities, which
allows for the possibility of a specific utility or disutility of gambling. It
is hoped that a way will be found to achieve this, but the mathematical
difficulties seem to be considerable. Of course, this makes the fulfillment
of the hope of a successful approach by purely verbal means appear even
more remote.

It will be clear from the above remarks, that the current method of using
indifference curves offers no help in the attempt to overcome these diffi-
culties. It merely broadens the concept of equality (c.f. the first remark
above), but it gives no useful indications — and a fortiori no specific instruc-
tions — as to how one should treat situations that involve probabilities,
which are inevitably associated with expected utilities.

INDEX OF FIGURES

Figure

Page

Figure

Page

Figure

Page

1

62

35

184

70

405

2

63

36

192

71

408

3

63

37

192

72

408

4

64

38

192

73

409

5

64

39

197

74

409

6

65

40

203

75

409

7

65

41

203

76

410

8

65

42

212

77

410

9

65

43

212

78

410

10

78

44

212

79

411

11

93

45

216

80

411

12

94

46

216

81

412

13

94

47

217

82

412

14

94

48

217

83

412

15

99

49

230

84

414

16

131

50

252

85

414

17

132

51

261

86

415

18

133

52

283

87

416

19

133

53

283

88

416

20

133

54

284

89

470

21

133

55

284

90

470

22

135

56

286

91

478

23

135

57

286

92

554

24

136

58

286

93

554

25

137

59

286

94

554

26

137

60

287

95

554

27

169

61

293

96

568

28a

175

62

295

97

568

28b

176

63

305

98

568

29

177

64

313

99

569

30

31

32

179

180

181

65

66

67

331

337

337

100

101

569

579

33

181

68

337

102

579

34

182

69

395

103

579

683

INDEX OF NAMES

Archimedes, 630 Kronecker, 129

Bernoulli, D., 28, 83, 629 Lipschitz, 493

Birkhoff, G., 62, 63, 64, 66, 340, 589 Loomis, L. H., vi

Bfihm-Bawerk. E. von. 9. 562. 564. 581. 582 MacLane. S.. 340

Bohr, N., 148

Bonessen, T., 128

Borel, E., 154, 186, 219

Brouwer, L. E. J., 154

Burnside, W., 256

Carathdodory, C., 343, 384, 619

Chevalley, C., vi

D'Abro, A., 148

Dedekind, 129, 619

Dirac, P. A. M., 148

Doyle, C., 176, 178

Euclid, 23

Fenchel, W., 128

Fr&nkel, A., 61, 595

Hausdorff, F., 61, 269, 595

Heisenberg, W., 148

Hilbert, D., 74, 76, 630

Hurwicz, L., vi

Kakutani, S., 154

Kaplanski, I., vi

Kepler, 4

Konig, D., 60

Marschak, J., vi
Mathewson, L. C., 256
Menger, C., 564
Menger, K., 28, 176
Mohs, 22

Morgenstern, O., 176, 178
Morse, M., 95

Neumann, J. von, 1, 154, 186

Newton, 4, 5, 6, 33

Pareto, V., 18, 23, 29

Speiser, A., 256

Tarski, A., 62

Tintner, G., 28

Tycho de Brahe, 4

Veblen, O., 76

Ville, J., 154, 186, 198

Wald, A., v, vi

Weierstrass, 129

Weyl, H., 76, 128, 256

Young, J. W., 76

Zermelo, E., 269, 595

634

INDEX OF SUBJECTS

Acyclicity, 589, 591, 594, 595, 596, 598,
600, 601, 602, 603, 609; strict, 594, 595,
597, 598, 600, 601, 602, 603
Additivity of value, 251, 628
Adversary “found out,” 105
Agreements, 221, 224; sanctity of, 223
Ally, 221

Alternatives, 55; number of, 69
Anteriority, 51, 52, 77, 78, 112, 117, 124
Apportionment, 35, 41, 504
Archimedean property, 630, 631
Assignment, actual, 75; pattern of, 75
Asymmetric, 270, 448
Austrian School, 9
Axiomatization, 68, 74, 76
Axioms, 25, 26, 28, 73; independence of,
76; logistic discussion of the, 76
Backgammon, 52, 58, 79, 124, 125, 164
Bargaining, 338, 501, 512, 557, 558, 572,
616

Barter exchange, 7

Behavior, 34; expected, 146; standards of,
see Standards of behavior
Best way of playing, 100
Bid, 557

Bidding, alternate, 211
Bilateral monopoly, 1, 6, 35, 504, 508, 543,
556

Bilinear form, 154, 156, 157, 166, 233
Bluffing, 54, 164, 168, 186, 188, 204, 205,
206, 208, 218, 541; fine structure of, 209
Boolean algebra, 62
Bound, 59, 60
Bounded, 384

Bounds, lower, 100; upper, 100
Bridge, 49, 52, 53, 58, 59, 79, 86, 224;

Duplicate, 113; Tournament, 113
Buyer, 14, 556, 557, 565, 569, 572, 574,
581, 583, 585, 609, 610, 613

Calculus, 3, 5, 6

Calculus of variations, 11, 95

Calling off, 179, 180, 541

Cartels, 15, 47

Categoricity, 76

Center of gravity, 21, 131, 303

Chance, 39, 52, 87

Characteristic function, 238 ff., 240, 245,
509, 510, 511, 527, 529, 530, 533, 535,
557, 574, 584, 605, 610; extended, 528,
529, 532, 533; game with a given, 243,

530, 532; interpretation, 538; in the
new theory, 348; normalized form of the,
325; reduced, 248, 325, 543, 544, 545;
restricted, 528, 529, 531, 532, 533;
strategically equivalent, 536; vector
operations on, 253; zero reduced, 545
Characteristic set function, 241
Chess, 49, 52, 58, 59, 113, 124, 125, 164
good, 125

Chief player, see Player, chief
Choice, 49, 51, 59, 69, 222, 508; actual, 75;
anterior, 72; pattern of, 75; umpire's, 80,
81, 82, 183

Choice, axiom of, 269
Circularity, 40, 42, 56
Closed set, 384

Coalition, 15, 35, 47, 221, 222, 224, 225,
229, 234, 237, 240, 260, 276, 289, 418,
420, 507, 509, 510, 531, 533, 539, 566,
572, 573, 605; absolute, 231, 238; cer-
tainly defeated, 440; certainly winning,
440; decisive, 420; defeated, 296; final,
315, 317; first, 306, 307, 315, 316, 320;
interplay of, 291; losing, 420, 421, 423;
minimal winning, 429, 430, 436, 438,
445; profitable minimal winning, 442;
unprofitable, 437; weighted majority,
434; winning, 296, 297, 333, 420, 421,
423, 436, 445, 470

Coalitions, competition for, 329; of differ-
ent strengths, 227
Closed systems, 400
Column of a matrix, 93, 141
Combinatorics, 45
Commodity, 10, 13, 560, 565
Communications, 86; imperfect, 86
Commutativity, 91ff., 94. See also Saddle
points

Compatible, 267 ff.

Compensations, 36, 44, 47, 225, 227, 233,
234, 235, 237, 240, 507, 508, 510, 511,
513, 533, 541, 558
Competition, 1, 13, 15, 249, 509
Complement, 62

Complementarity, 18, 27, 251, 437, 628
Complementation, 422
Complete ordering, see Ordering, complete
Completely defeated, 296
Composition, 340, 359, 360, 454, 548; of
simple games, 455

635

636

INDEX OF SUBJECTS

Conjunction, 66

Constituent, 340, 353, 359, 360, 518;
indecomposable, 457, 471; inessential,
•453, 457; simple, 453, 455, 457
Contribution, 364, 366
Conventions, 224
Convex bodies, 128
Convexity, 128 ff., 275, 547
Cooperation, 221, 402, 474, 481, 508, 517;

complete, 483
Couple, 222, 226, 243, 509
Crusoe, 9, 15, 31, 87, 555
Cube Q t 293, 295; center and its environs,
313; center of, 316, 317, 321 ; corner, 303,
304, 307, 340, 429; interior of, 302, 303,
304; main diagonal of, 302, 304, 305, 312;
neighborhood of the center, 321; special
points in, 295 ff. ; three-dimensional part
of, 314

Curves, undetermined, 418
Cutting the deck, 185, 186
“Cyclical” dominations, 39

Decomposability, 342, 357, 360; analysis
of, 343

Decomposition, 242, 292, 340, 359, 360,
452, 537, 548; elementary properties of,
381 ; its relation to the solutions, 384
Decomposition Partition, see Partition,
decomposition

Defeated, 296; certainly, 440; fully, 436.

See also Players; Coalitions
Defensive, 164, 205

Determinateness, general strict, 150, 155,
158 ff.; special strict, 150, 155; strict,
106 ff., Ill ff., 165, 178, 179
Diagonals, separation of the, 173
Differential equations, 6, 45
Direct signaling, 54

Discrimination, 30, 288, 289, 328, 475, 476,
512. See also Solution, discriminatory
Disjunction, 66
Distance, 20

Distribution, 37, 87, 226, 261, 263, 350,
364, 437

Domain, 89, 90, 128, 157
Domination, 38, 264, 272, 350, 367, 371,
415, 474, 520 ff., 522, 523, 524, 587;
acyclical concept of, 602; asymmetrical,
270; extension of the concept of, 587;
intransitive nation of, 37
Double-blind Chess, 58, 72, 79
Duality, 104

Dummy, 299, 301, 340, 358, 397, 398, 400,
454, 455, 457, 460, 461, 508, 518, 537,
538

Duopoly, 1, 13, 543, 603
Dynamic equilibria, 45
Dynamics, 44, 45, 189, 290

E(e 0 ), Solutions for r in, 393 ff.

Ecart6, 59

Economic equilibrium, 4; fluctuations, 5;

models, 12, 58; statics, 8
Economics, mathematical, 154
Economies, internal, 341
Economy, planned communistic, 555;
Robinson Crusoe, 9; Social Exchange,
9 ff.

Effectivity, 272, 282, 350, 367, 524
Energy, 21
Entrepreneur, 8
Equidistributed, 197
Equilibrium, 4, 34, 45, 227, 365
Equivalence strategic, see Strategic equiv-
alence

Essentiality, 249, 272, 351, 452
Exceptional, 593

Excess, 364, 367, 417, 418, 454, 455, 548;
distribution of the, 418; limitation of
the, 365, 366; lower limit of, 368; too
great, 374, 380, 419; too small, 374, 380;
upper limit of, 369
Exchange economy, 9, 31
Exchange, indeterminateness of, 14
Excluded player, see Player, excluded
Expectation, 12, 28, 83, 539; mathematical,
10, 28, 29, 32, 33, 83, 87, 117, 118, 126,
149, 156, 157; moral, 28, 83; values, 183
Exploitation, 30, 329, 375
Extensive form, 112, 119

F(e 0 ), Solutions for r in, 384 ff.

Fairness, 166, 167, 225, 255, 258, 259, 470
Fictitious player, see Player, fictitious
“Finding out” the other player, 148
“Finds out” his adversary, 106, 110, 148
First element, 38, 271
Fixed payments, 246, 281, 298, 534
Fixed Point Theorem, 154
Flatness, 276, 547
Found out, 148
Frame of reference, 129
Fully detached, see Imputation, detached
fully

Function, 88, 128; arithmetical, 89; char-
acteristic, 238 ff.; continuity of, 493;
measure, 252; numerical, 89; numerical
set, 240, 243, 530, 532; of functions,
157; set, 89
Functional, 157
Functional Calculus, 88, 154
Functional operations, 88, 91
Fundamental triangle, 284, 405 ff., 552,
553, 569, 570, 587; area, 579, 580;
curves in, 412, 570, 580; undominated
area, 409

Gain, 33, 128, 145, 539, 556, 559, 629
Gambling, 27, 28, 87, 630, 631

INDEX OF SUBJECTS

637

Game and social organizations, 43; asym-
metric, 334; auxiliary, 101 ff.; axiomatic
definition of a, 73; chance component of
the, 80; classification of, 46; complete
concept of, 66; complete system of rules
of, 83; composition, 339 ff.; constant-
sum, 346 ff., 347, 360, 361, 504, 605, 535,
536, 537, 586; decomposable, 454, 471,
518; decomposable, solution of, 358, 381 ;
decomposition, 339 ff.; direct majority,
431, 433; elements of the, 49; essential,
231, 232, 245, 250, 331, 534, 546;
essential three-person, 220 ff., 260 ff.,
471, 473; everyday concept of, 32;
extensive form of the, 85, 105, 186, 234;
extreme, 534, 535; fictitious, 240; final
simplification of the description of a,
79, 81; general, 504 ff., 505, 538; general
description of, 57; general formal
description of, 46-84; general n-person,
48, 85, 112, 530, 606; general n-person,
application of theory, 542 ff . ; imbedding
of a, 398; indecomposable, 354; ines-
sential, 231, 232, 245, 249, 251, 471;
“inflation ” of a, 398; invariant, 257;
kernel of, 457, 459; length of the,
75; main simple solution of the, 444;
majorant, 100, 102, 103, 119, 149;
majority and the main solutions, 431;
minimum length of, 123; minorant, 100,
101, 119, 149; non-isolated character
of a, 366; non-strictly determined, 110;
non-zero-sum, 47. See also Game, gen-
eral; normal zone of the, 519; normalized
form, 85, 100, 105, 119, 183, 234, 239,
322, 325, 452, 473; of chance, 87, 185;
one-person, 85, 548; partitions which
describe a, 67; perfect information,
112 ff.; plan of the, 98; plays of, 49;
reduced, 248, 259, 473, 543 ff.; rules of
the, 32, 49, 59, 80, 113, 147, 224, 226,
227, 334, 426, 472; set-theoretical

description of a, 60, 67; simple, see
Simple game; simplified concept of a,
48; strategies in the extensive form, 111 ;
strictly determined, 98 ff., 106, 124, 150,
165, 172, 174, 516; strictly determined,
generally, 150; strictly determined,
specially, 150; struggle in, 125; super-
position of, 254, 255; surprise in, 125;
symmetric, 165, 167, 192, 195, 334, 362;
symmetric five-person, 332, 334; sym-
metry, total of, 259; three-person, 35,
220 ff., 282 ff., 403 ff., 457, 550; three-
person, simple majority of, 222 ff . ;
totally symmetric, 257; totally unsym-
metric, 257; unique, 331; vacuous, 116,
546; value of the, 102, 103, 170, 516;
weighted majority, homogeneous, 444;
zero-sum extension of, 529; zero-sum

four-person, 291 ff.; zero-sum n-person,
48, 85, 238 ff.; zero-sum three-person,
220 ff., 260 ff.; zero-sum three-person,
solution of essential, 282 ff.; zero-sum
two-person, 48, 85 ff., 116, 169 ff., 176
Geometry, 20, 74, 76; linear, 428; plane,
7-point, 469; projective, 469
Good way (strategy), 103
Good way to play, 108, 159
Goods, complementary, 437, 628; divisible,
560, 573

Group, 22, 76, 255 ff.; alternating, 258;
invariance, 257; of permutations, 256;
set-transitive, 258; symmetric, 256;
theory, 256, 258, 295; totally symmetric,
257; totally unsymmetric, 257
Group (of players), privileged, 320

Half-space, 130, 137, 139
Hand, 53, 186, 187, 190, 197, 613
Hands, discrete, 208
Heat, 3, 17, 21
Hereditary, 454
Heredity, 396, 400

Heuristic, 7, 25, 33, 120, 238, 263, 291,
296, 298, 301, 302, 307, 316, 318, 322,
333, 499, 509, 511, 587
Heuristic argument, 147, 181, 182, 227
Higgling, 557, 558
Homo oeconomicus, 228
Homogeneity, 433, 464
Hyperplane, 130, 137, 139; supporting,
134 ff.

Identity, 255, 617
Imbedding, 398, 399, 400, 455, 587
Imputation, 34, 37, 39, 240, 251, 263, 264,
350, 376, 437, 517, 520, 527, 566, 577,
587, 606, 610; composition of, 359;
decomposition of, 359; detached, 369,
370, 375 ff., 413; detached extended,
370, 375; detached extended, fully, 370,
372; detached, fully, 369, 413; economic
concept of, 435; extended, 364 ff., 367,
368, 369, 372; single, 34, 36, 37, 39, 40;
unique, 608

Imputations, extended, sets of, 368; finite
set of, 465, 499; infinite set of, 288, 499;
isomorphism between, 282; set of, 34,
44, 608; system of, 36, 277, 464
Incomparable, 590, 630
Indecomposability, 357 L630, 631, 632]

Indifference curves, 9, 16, 19, 20, 27, 29,
Indifference of player, 300
Indirect proof, 147
Individual planning, 15
Induction, 112, 116; complete, 113, 123;

finite, 597; transfinite, 598
Inessential games, 44

638

INDEX OF SUBJECTS

Inessentiality, 249, 272, 351, 357, 454
Information, 47, 51, 54, 55, 56, 58, 67, 71,
109, 112, 183; absurd, 67; actual, 67, 79;
chance, 182; complete, 30, 541, 582;
imperfect, 30, 182; incomplete, 30, 86;
pattern of, 67, 69; perfect, 51, 123, 124,
164, 233; perfect, verbal discussion, 126;
player’s actual, 75; player’s pattern of,
75; sets of, 77; umpire’s actual, 75;
umpire’s pattern of, 75, 77; umpire’s
state of, 115
Initiative, 189, 190
Inner triangle, 409, 413, 553
Interaction, 341, 366, 400, 483
International trade, 7, 341
Interval, linear, 131
Intransitivity, 39, 52
Inverted signaling, 54
Irrational, 128, 523
Isomorphic correspondence, 281
Isomorphism, 149, 350, 504
Isomorphism proof, 281

Just dues, 360, 361

Kernel, 457

Killing the variable, 91

“Kriegsspiel,” 58

Laisser faire, 225
“Lausanne” Theory, 15
Linear interpolation, 157
Linear transformation, 22, 23
Linearity, 128 ff.

Lipschitz condition, 493
Logic, 62, 66, 74, 274
Logistic, 66
Losing, 421, 426

Loss, 33, 128, 145, 163, 167, 168, 205, 539,
555, 559, 629

Main condition, 273, 274, 279
Main simple solution, see Solution, main
simple

Main theorem, 153
Majorities, weighted, 432
Majority, principle of, 431
Majority game, homogeneous, 443; homo-
geneous weighted, 444, 463, 469;

weighted, 433, 464

Majority principle, homogeneous weighted,
467", 469

Majorization of rows or columns, 174, 180
Mapping, 22, 618 ff.

Marginal pairs, 560, 562, 563, 564, 571, 581
Marginal utility school, 7
Market, 47, 504, 556, 557, 560, 563, 581,
605; general, 583; three-person, 564;
two-person, 555
Mass, 20, 21

Matching Pennies, 111, 143, 144, 164, 166,
169, 176, 178, 185; generalized forms,
175 ff.

Mathematical Method, difficulties of, 2;

in Economics, 1-8
Mathematical physics, 303
Matrix, arbitrary, 153; diagonal of, 173,
174; elements, 93, 138, 141; negative
transposed, 141, 142; rectangular, 93,
138, 140, 141; scheme, diagonal of the,
173; skew symmetric, 142, 143, 167
Max operations, 89 ff.

Maximum, 88, 89, 591, 593, 594; absolute,
591, 593; collective benefit, 513, 514,
541, 613; problem, 10, 11, 13, 42, 86, 87,
220, 504, 517, 555; relative, 592; satis-
faction, 10

Measurability, 16, 343
Measure, additive, 343; mathematical
theory of, 252, 343
Measurement, principles of, 16, 20
Mechanics, 4

Method, mathematical, 322
Method, saturation, 446
Min operations, 89 ff.

Min- Max problem, 154
Minimal elements of W f 430, 448
Minimum, 88, 89
Mistake, 162, 164, 205
Mixing the deck, 185

Models, axiomatic, 74; economic, 12, 58;

mathematical, 21 ff., 32, 43, 74
Money, 8, 10
Monopolist, 474

Monopoly, 13, 474, 543, 584, 586, 602, 603
Monopsony, 584, 586, 602, 603
Monotone transformations, 23
Motivation, 43

Move, chance, 50, 69, 75, 80, 83, 112, 118,
122, 124, 126, 183, 185, 190, 517, 604;
removing of, 183; dummy, 127; impos-
sible, 72; in a game, 49, 55, 58, 59, 72,
98, 109, 111; of the first kind, 50; of
the second kind, 50; personal, 50, 55,
70, 75, 112, 122, 126, 183, 185, 190, 223,
508, 510

Negation, 66

Negotiations, 263, 534, 541
New Theory, characteristic function in the,
348; decomposability in the, 351; domi-
nations in the, 350; essential three-
person game in the, 403 ff.; essentiality
in the, 351; imputations in the, 350;
unessentiality in the, 351; solutions in
the, 350

Normal zone, 396, 399, 401, 417
Numerical utilities axiomatic treatment of,
24. See also Utility, numerical

INDEX OF SUBJECTS

639

Numerical utility, 17. See aho Utility,
numerical

Numerical weight, 432

Offensive, 164, 205
Oligopoly, 1, 13, 47, 504
Opposition of interest, 11, 220, 484
Optimality, permanent, 162, 205. See also
Strategy
Optimum, 38
Optimum Behavior, 34
Order of society, 41, 43. See aho Organi-
sation, Standard of behavior
Ordering, 19, 37, 38, 589; complete, 19, 26,
28, 589, 591, 593, 595, 600, 617; partial,
590, 591, 600; well, 595
Oreographical, 95, 97
Organisation, 224, 328, 366, 401, 419;
social and economic, 41, 43, 225, 318,
319, 329, 358, 362, 365, 402, 436, 471;
social, complexity of, 466
Origin (point in space), 129
Outside source, 363, 364, 366, 375, 419
Overbid, 186. See aho Poker

Parallelism of interests, 11, 220, 221
Partial ordering, 590. See also Ordering
Participants, 31, 33

Partition, 60, 63, 64, 66, 67, 69, 84, 114 ff.;
decomposition, 353, 354, 356, 357, 457,
471; logistic interpretation, 66
Pass, 95, 97
Passing, see Poker
Patience, 86

Per absurdum proof, 147, 148
Permutation, 255, 262, 294, 309, 319;

cyclic, 230, 470
Perturbations, 303, 341
Physical sciences, 21, 23, 32, 43
Physics, 2, 3, 4, 45, 76, 148, 401
Planning, 86
Plateau, rectangular, 97
Plausibility considerations, 7
Play, 49, 59, 71; actual, 82; course of the,
68, 70, 84; individual identity of the, 71 ;
outcome of the, 82; sequence of choices,
49; value of a, 104, 105, 124, 127, 150,
163, 165, 178, 238

Player, chief, 473, 474 ff., 481, 483, 500,
502; chief, segregated, 500, 502; com-
posite, 231, 232, 239, 516; defeated, 418;
discriminated, 476, 502; excluded, 289,
301, 512; fictitious, 505, 506, 507, 508,
509, 511, 513, 514, 516, 518, 537; found
out, 146; indifferent, 299; isolated, 375;
privileged, 473; segregated, 476, 502,
503; self-contained, 353, 357; splitting
the, 86; unprivileged, 320; victorious,
426

Players, interchanging the, 104, 109, 122,
165, 255; permutation of the, 294, 463;
privileged group of, 320, 464; removable
sets of, 533; strategies of the, 49. See
also Strategy.

Playing appropriately, 102, 103, 107, 159,
167

Plays, set of all, 75

Poker, 52, 54, 56, 58, 59, 164, 168, 186,
187 ff., 557, 613; bids, 209; Draw, 187;
general forms of, 207; good strategy,
196; interpretation of the solutions, 218;
mathematical description of all solu-
tions, 216 ff.; overbidding, 188, 190;
passing, 188, 190, 191, 199; seeing, 188,
190, 191, 199, 218; solution, 199, 202,
204; strategies, 191; Stud, 187
Position, 21
Positive octant, 133
Positive quadrant, 133
Postulates, 27. See aho Axioms
Preferences, 15, 17, 18, 23, 522, 590, 607,
630; completeness of, 29, 630, 631;
transitivity of, 27. See aho Utility
Preliminarity, 51, 52, 77, 78, 112, 117, 124
Preliminary condition, 273, 471
Premiums, 582

Price, 556, 559, 562, 563, 564, 571, 572,
582, 585; average, 564, 582; unique, 564
Privilege, 464

Probabilities, choice of, 145
Probability, 11, 17, 19, 39, 81, 87, 128, 146,
197 ; geometrical, 197; numerical, 14, 19,
27, 28, 69, 75, 80, 113, 145, 147, 156, 183,
604; of losing, 144; of winning, 144
Production, 5, 13, 504
Productivity, 33, 34, 504, 540
Profit, 33, 47, 572
Proper subset, 61
Proper superset, 61

Psychological phenomena, mathematical
treatment, 28, 77, 169

Quantum mechanics, 3, 33, 148, 401

Randomness, 146

Rational, 9, 517; behavior, 8-15, 31, 33,
127, 150, 160, 224, 225; playing, 54
Rationality, 99, 128
Rebates, 582
Recontracting, 557, 558
Reduced form, 248, 544
Reduction, 322, 325
Relativity Theory, 23, 148
Removable set, see Set, removable
Ring, 243, 530, 531
Risk, 163

Robinson Crusoe, see Crusoe

640

INDEX OF SUBJECTS

Rolling dice, 166
Roulette, 87

Row of a matrix, 93, 141. See also Matrix

Saddle, 95, 97; points, 93, 95, 110, 153;

value, 88, 95, 107
St. Petersburg Problem, 28, 83
Satisfaction, maximum, 8, 10, 15
Satisfactoriness, 267 ff., 446 ff.; maximal,
269

Saturation, 266, 267 ff., 446 ff., 448, 591
Scalar multiplication, 129, 253, 254
Seeing, see Poker
Segregation, 290

Seller, 556, 557, 565, 569, 572, 574, 581,
583, 585, 609, 610
Separated numbers, 173
Set, 60, 61, 114 ff.; certainly necessary,

273, 274, 277, 308, 309, 323, 405, 430,
471, 547; certainly unnecessary, 273,

274, 276, 308, 309, 323, 405, 430, 471,
547; completely ordered, 19; convex,
131, 133; convex spanned, 131; effective,
38, 264; elements of, 61; empty, 61, 241,
380; finite, 61; flat, 275, 276, 423, 424;
minuend, 62; one-element, 61; partially
ordered, 19; removable, 533, 534, 535;
solo, see Solo set; splitting, 353, 457, 518;
splitting minimal, 355, 356; splitting,
system, of all, 354; subtrahend, 62;
theory, 45, 60 ff.

Sets, difference of, 62; disjunct, 62; inter-
section of, 62; logistic interpretation, 66;
of imputations, composition of, 359; of
imputations, decomposition of, 359;
pairwise disjunct, 62; product of, 62, 66;
self-contained, 354; sum of, 62, 66; sys-
tems or aggregates of, 61
Signaling, 51, 53; direct, 54; inverted, 54
Simple game, 420 ff., 454, 605; adding
dummies to, 462; and decomposition,
452; characteristic function, 427; char-
acterization of the, 423; complementa-
tion in, 422 ff.; enumeration of all,
445 ff.; for n ^ 4, 461; for n 6, 463;
for small n, 457; indecomposable, 457;
one-element sets in the, 425 ff. ; six main
counter-examples, 464; solution of, 430;
strategic equivalence, 428; systems W,
L , of, 426 ff.; with dummies, 461
Simplicity, 433, 452, 454; elementary
properties of, 428; exact definition of,
428

Skew-symmetry, 166
Social exchange economy, 9 ff .

Social order, 365

Social organizations, see Organizations,
social

Social product, 46
Social structure, 484
Society, 42, 320, 341, 523, 540
Solitaire, 86
Solo set, 244, 530, 531
Solution, 102, 264, 350, 367, 368, 417, 478,
526, 527, 587, 588; areas (two-dimen-
sional parts) in the, 418; asymmetric,
315, 362; composition of, 361; concept
of a, 36; concept of imputation in form-
ing, 435; curves (one-dimensional parts)
in the, 417; decomposable, 362; decom-
position, 361; definition, 39, 264; dis-
criminatory, 301, 307, 318, 320, 329,
442, 511, 512; essential zero-sum three-
person game, 282; existence of, 42;
extension of the concept of, 587 ;
families of, 329, 603; finite, 307, 500;
finite sets of imputations, 328; for an
acyclic relation, 597; for a complete
ordering, 591; for r in E(e 0 ), 393 ff.;
for r in F(e 0 ), 384 ff.; for a partial
ordering, 592; for a symmetric relation,
591; general games with n ^ 3, of all,
548; general three-person game, 551;
indecomposable, 362; inobjective (dis-
criminatory), 290; main simple, 444,
464, 467, 469; multiplicity of, 266, 288;
natural, 465; new definition of, 526;
non-discriminatory, 290, 475, 511; objec-
tive (non-discriminatory), 290; one-
element, 277, 280; set of all, 44; super-
numerary, 288; symmetric, 315; unique,
594, 600, 601, 603; unsyrametrical

central, 319
Soundness, 265

Space, Euclidean, 21, 128, 129; half, 137 ;
linear, 157; n-dimensional linear, 128;
positive vector, 254
Special form of dependence, 56
Splitting the personality, 53
Stability, 36, 261, 263, 266, 365; inner, 42,
43, 265

Standard of Behavior, 31, 40, 41, 42, 44,
265, 266, 271, 289, 361, 365, 401, 418,
472, 478, 501, 512, 513; discriminatory,
290; multiplicity (of stable, or accepted),
42, 44, 417; non-discriminatory, 290
Statics, 44, 45, 147, 189, 290
Statistics, 10, 12, 14, 144
Stone, Paper, Scissors, 111, 143, 144, 164.
185

Stop rule, 59, 60

Strategic equivalence, 245, 247, 248, 272,
281, 346, 348, 373, 426, 429, 472, 505,
535, 543; isomorphism of, 504, 505
Strategies, 44, 50, 79, 80, 84, 101, 117, 119;
combination of, 159

INDEX OF SUBJECTS

641

Strategy, as move, 84; asymptotic, 210;
best, 124, 517; choice of, 82, 145, 147;
concept of a, 79; found out, 151, 153,
158, 160, 168; good, 108, 146, 160 ff.,
161, 162, 164, 170, 172, 178, 179, 183,
196, 205, 206; higher order of a, 84;
mixed, 143 ff., 146, 148, 149, 155, 157,
161, 168, 174, 183, 192, 232, 539, 604;
optimal, 127, 517; permanently optimal,
163, 164, 165; pure, 146, 148, 155, 157,
161, 168, 181, 182; statistical, 144, 146;
strict, 146; structure, continuous, 197;
structure, fine, 197.; structure, granular,
197

Strict Determinateness, see Determinate-
ness

Struggle, 249
Subpartition, 63, 64, 69
Subset, 61

Substitution rate, 465
Superiority, intransitive notion of, 37
Superposition, 64
Superset, 61

Symmetry, 104, 109, 165, 166, 190, 224,
255, 256, 258, 267, 315, 446 ff., 500, 591.
See also Group

Tautological, 8, 40
Temperature, 17, 21
Theory, extended, structure of the, 368
Theory, new, 526, 528. See also New
theory

Thermodynamics, 23

Thermometry, 22

Tie, 125, 315

Topology, 154; 384

Total value, 251

Transfer, 30, 364, 365, 401, 402

Transferability of utility, 8, 608

Transfinite induction, 269

Transformation, 22, 23

Transitivity, 38, 39, 51, 589, 590

Trees, 66, 67

Tribute, 30, 402

Tug-of-war, 100

Umpire, 69, 72, 84
Uncertainty, 35

Understandings, 223, 224, 237
Utilities, comparability of, 29; complete
ordering of the, 19, 26, 29, 604, 617 ff. ;
differences of, 18, 631; domain of, 23,.
607; nonadditive, 250; non-numerical,
16, 606, 607; numerical, 17 ff., 157,

605, 606, 617 ff.; numerical, substitut-
ability, 604; partially ordered, 19, 590;
system of, 26; transferability, 8, 604,

606, 608, 629; variable, 560

Utility, 8, 15, 23, 33, 47, 83, 156, 556, 563,
565, 569, 572, 573, 583, 585, 603, 608,
616, 617 ff. ; axiomatic treatment, 26 ff.,
617 ff.; decreasing, 561, 576; discrete,
613; expected, 30; generalization of the
concept of, 603 ff . ; indivisible units, 609,
613, 614; marginal, 29, 30, 31; scale,
fineness of, 616; total, 34, 35

Value, economic, 252, 467, 556, 565; of a
function, 88; of a play, see Play, value of
Variables, 12, 13, 88; aggregate of, 239;

“alien,” 11; partial sets of, 12 ff.

Vector, 129, 140; addition, 130, 253, 254;
components, 129, 404; coordinate, 129,
157; distance, 134; length, 134; oper-
ations, 129; quasi-components, 404;
spaces, 254; zero, 129
Victorious, 296. See also Player; Coali-
tion; Winning; Losing
Virtual existence, 36, 45, 338, 484

Wants, 10

Wave mechanics, 148
Weights, 433, 434, 463; homogeneous, 435,
444

Who finds out whom, 110
Winning, 296, 421, 426; certainly, 440;
fully, 436

Withdrawal, 364, 366

Zero-reduced form of characteristic func-
tion, 545

Zero-sum condition, 345
Zero-sum extension of T, 505, 506, 527,
531, 538

Zero-sum restriction, 84, 504